diff --git a/-NAzT4oBgHgl3EQfSvt8/vector_store/index.pkl b/-NAzT4oBgHgl3EQfSvt8/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b8869ce111b82083049b7d23d7ee28c38cea78b3 --- /dev/null +++ b/-NAzT4oBgHgl3EQfSvt8/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8537c828041cfaa690de7b0dc6a892c8c05aa9a9bfe07d80cd6e4edb4ab20540 +size 159975 diff --git a/-NFPT4oBgHgl3EQfZTSq/content/tmp_files/2301.13077v1.pdf.txt b/-NFPT4oBgHgl3EQfZTSq/content/tmp_files/2301.13077v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ee39d809ddc73a3b084cd15db3e8fd4e57e05e8 --- /dev/null +++ b/-NFPT4oBgHgl3EQfZTSq/content/tmp_files/2301.13077v1.pdf.txt @@ -0,0 +1,1113 @@ +A sluggish random walk with subdiffusive spread +Aniket Zodage1,2, Rosalind J. Allen3,4, Martin R. Evans4,5, Satya N. Majumdar5 +1 Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India +2 Department of Physics, UC San Diego, 9500 Gilman Dr. La Jolla, California 92093, USA +3 Theoretical Microbial Ecology, Institute of Microbiology, Faculty of Biological Sciences, +Friedrich Schiller University Jena, Buchaer Strasse 6, 07745 Jena, Germany +4 SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, EH9 3FD and +5 LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France +(Dated: January 31, 2023) +We study a one-dimensional sluggish random walk with space-dependent transition probabilities. +Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases +logarithmically with distance from the origin. This leads to a random walk which has symmetric +transition probabilities that decrease with distance |k| from the origin as 1/|k| for large |k|. We show +that the typical position after time t scales as t1/3 with a nontrivial scaling function for the position +distribution which has a trough (a cusp singularity) at the origin. Therefore biased random motion +emerges even though the transition probabilities are symmetric. +We also compute the survival +probability of the walker in the presence of a sink at the origin and show that it decays as t−1/3 at +late times. Furthermore we compute the distribution of the maximum position, M(t), to the right +of the origin up to time t, and show that it has a nontrivial scaling function. Finally we provide a +generalisation of this model where the transition probabilities decay as 1/|k|α with α > 0. +arXiv:2301.13077v1 [cond-mat.stat-mech] 30 Jan 2023 + +2 +I. +INTRODUCTION +Slow dynamics is a common feature of many physical systems, including glasses, granular media and colloids [1, 2]. +well Slow dynamics commonly arises when the system becomes trapped for increasing periods of time in deeper and +deeper local free energy minima in the configuration space. This phenomenon has inspired the study of simplified +toy models known as trap models [3–7]. In these models, the many minima of the complex disordered landscape are +represented by traps whose depths are taken to be random variables. For a single particle hopping between nearby +traps the mean squared displacement typically grows more slowly than linearly in time, thus the particle’s motion is +subdiffusive [8–10]. +Similar slow dynamics can also arise in an inhomogeneous landscape where the trap depth is position-dependent +(as opposed to being random). Here, the hopping dynamics between the traps is an example of a Markov chain +with space-dependent transition probabilities [11, 12], or in other words, an inhomogeneous random walk. For such +systems, explicit solutions for observables beyond the simple position distribution – such as first passage probabilities +[13–15] or extreme value statistics [16, 17] – are generally hard to obtain. +A classic example of an inhomogenous random walk is the centrally-biased Gillis model [18–22]. In this model, a +single particle hops on a one-dimensional lattice where the hopping probability is asymmetric in a position-dependent +manner. Specifically, for k ̸= 0, the hopping probabilities from site k to k ± 1 are 1 +2(1 ∓ ϵ/k), while for k = 0 the +hopping probabilities to sites ±1 are both 1/2. Because the hopping probabilities (for k ̸= 0) are asymmetric, the +particle undergoes a biased random walk in which the parameter ϵ ∈ [−1, 1] controls the strength of the bias. For +ϵ > 0 there is a drift towards the origin while for ϵ < 0 there is a drift away from the origin. Far away from the origin, +where |k| ≫ 1, the bias is small and the dynamics tends towards a symmetric random walk which, in the continuum +limit, reduces to a particle moving in a logarithmic potential U(k) → 2ϵ ln |k| [19]. +The Gillis model [18–22], and its continuous limit of particle motion in a logarithmic potential [19, 23–27], have +aroused much interest because of their relevance to vortex dynamics, interactions between tracer particles in a driven +fluid, cold atoms trapped in optical lattices and the nonequilibrium behaviour of systems with long-range interactions +[28–32]. These models have the appealing feature of allowing the exact calculation of various observables going beyond +the position distribution. +Motivated by these works, here we consider the counterpart problem of an inhomogeneous trap model in which +the trap depth increases logarithmically with increasing distance from the origin. The dynamics of a particle in this +model corresponds to a symmetric random walk with space-dependent hopping probabilities that decrease inversely +with distance from the origin. This random walk has the interesting property of being ‘sluggish’ since the particle’s +motion slows down as it goes further away from the origin. We show that the physics of this model is quite different +from the previously studied case of a particle in a logarithmic potential. Instead, in the continuum limit and for +|k| ≫ 1, our model corresponds to a particle moving in a potential U(k) ∼ 1/|k| with, additionally, a space-dependent +diffusion constant that also decays as 1/|k|. The interplay of these two features leads to an emergent bias in the +dynamics away from the origin, even though the hopping probabilities are symmetric. The position distribution has +a non-trivial and non- Gaussian form in which distance scales with time as t1/3 at late times. Moreover, we show +that other observables such as the survival probability in the presence of an absorbing site and the distribution of the +maximum displacement to one side of the origin can be computed explicitly and exhibit non-trivial scaling behaviour. +Finally we discuss how the model can be easily generalised to higher dimensions and other space-dependent hopping +probabilities, such as a 1/|k|α decay with exponent α > 0, without losing its solvability. +II. +MODEL DEFINITION AND HYDRODYNAMIC LIMIT +As discussed, we consider an ordered array of traps arranged on a one-dimensional lattice such that the depth of +the trap at site k is a ln(|k| + 2) (as illustrated in Fig. 1). The corresponding Arrhenius escape rate from the trap at +site k is A(|k| + 2)−α with α = β a, where A is an overall constant and β is the inverse temperature. Without loss +of generality we will set A = 1. We will mostly focus on the case where α = 1, although we briefly discuss α ̸= 1 in +Section IX. +We consider the discrete-time dynamics of a particle moving at random on this infinite one-dimensional lattice. +The key feature of the dynamics is that, as time progresses, the particle explores sites further and further away from +the origin in which it gets trapped to a greater and greater extent because of the increasing trap depths. Hence the +particle is subject to diminishing transition probability for exiting the traps. +At each integer time step t the particle’s position evolves according to the following rules (illustrated in Fig. 1). +From a site k at time t, the particle hops to site k + 1 with probability 1/(|k| + 2), it hops to site k − 1 with equal +probability 1/(|k|+2), or it stays at site k with the complementary probability |k|/(|k|+2). The time is then updated +to t + 1. We note that (in contrast to the Gillis model [18–22]) the hopping probabilities are symmetric for all k; + +3 +FIG. 1. Schematic illustration of our model, showing the ordered arrangement of traps and the hopping probabilities 1/(|k|+2) +for a particle to move to a nearest neighbour of site k on the lattice, as well as the probability |k|/(|k| + 2) of remaining at site +k. +however they differ from those of a simple random walk everywhere except at the origin, k = 0, where the hopping +probability is 1/2 to either of k = ±1. +Let P(k, t) denote the position distribution at time t for a particle that starts from k0 = 0 at t = 0. The distribution +evolves via the forward master equation: +P(k, t + 1) = +1 +|k + 1| + 2 P(k + 1, t) + +1 +|k − 1| + 2 P(k − 1, t) + +|k| +|k| + 2 P(k, t) . +(1) +The initial condition is P(k, 0) = δk,0 and the boundary condition is P(k, t) → 0 as |k| → ∞. The solution P(k, t) is +symmetric around k = 0, hence we can just focus on k ≥ 0. We first write (1) in in the more suggestive form +P(k, t + 1) − P(k, t) = +1 +|k + 1| + 2 P(k + 1, t) + +1 +|k − 1| + 2 P(k − 1, t) − +2 +|k| + 2 P(k, t) . +(2) +For large k > 0 and large t, we can expand the right hand side (rhs) of Equation (2) as a Taylor series in k and +replace the left hand side (lhs) by a time derivative. This gives, keeping all terms of the same order, the hydrodynamic +equation that captures the behaviour of the system at long distance and at late time: +∂ +∂tP(k, t) ≈ 1 +k +� ∂2 +∂k2 P(k, t) − 2 +k +∂ +∂k P(k, t) + 2 +k2 P(k, t) +� +. +(3) +This hydrodynamic equation can be written as a continuity equation: +∂ +∂tP(k, t) = − ∂ +∂k j(k, t) , +(4) +where the space-time dependent current density j(k, t) reads +j(k, t) = −1 +k +∂ +∂k P(k, t) + 1 +k2 P(k, t) . +(5) +We identify the first term on the rhs of (5) as a diffusive probability current and the second term as a drift away from +the origin. The original equation (3) can also be expressed in the standard Fokker-Planck form +∂ +∂tP(k, t) = ∂ +∂k +� +D(k) ∂ +∂k P(k, t) + +� ∂ +∂k U(k) +� +P(k, t) +� +, +(6) +where D(k) = 1/k and U(k) = 1/k. Equation (6) shows clearly the two features of the dynamics that lead to non- +trivial behaviour. Firstly, the effective diffusion constant D(k) = 1/k is space dependent, such that it slows down + +4 +the dynamics as the particle moves away from the origin. Additionally, an effective external potential U(k) = 1/k +emerges, which is repulsive, such that it pushes the particle away from the origin. This repulsion arises from the +microscopic dynamics: the hopping probability from site k to k + 1 is 1/(k + 2), while the reverse event (from (k + 1) +to k) has probability 1/(k + 3) < 1/(k + 2) for any k > 0. Thus, even though the hopping probabilities out of site +k (i.e. to (k + 1) or (k − 1)) are symmetric, the space-dependence of the hopping probability produces an outward +bias away from the origin (symmetrically for k < 0) which leads to the drift term in the current in equation (5) in +the hydrodynamic description. +For large time and space we expect to see a scaling regime in which the probability distribution becomes a function +of a combination k/tν (with ν > 0). The following argument suggests that ν takes the value 1/3. Let us assume that +after time t, the typical value of the position k is ktyp. The number of steps N that have been taken by the particle +will scale as N ∼ t/ktyp, since the time for one step is the typical escape time 1/ktyp. Since all steps are equal in +distance and the hopping probability is symmetric, the position scales with the number of steps in the same way as +for a simple random walk, ktyp ∼ N 1/2. Putting these arguments together we obtain ktyp ∼ N 1/2 ∼ (t/ktyp)1/2 which +implies that the position of the particle scales with time as ktyp ∼ t1/3; hence ν = 1/3. +This scaling can be confirmed by assuming the following scaling form for the probability distribution P(k, t) in the +limit when both k and t are large, keeping the ratio k/tν (with ν > 0) fixed: +P(k, t) → +1 +b tν G +� k +b tν +� +, +(7) +where G(z) is the scaling function. We have also incorporated an adjustable constant b which can be chosen appro- +priately. Substituting the scaling form (7) in equation (1), one readily finds that for leading order terms to be of the +same order we must have ν = 1 +3. For convenience we will also choose b = 31/3. We will discuss the precise form of +the scaling function G(z) in Section IV. +The scaling k ∼ t1/3 also manifests itself in other observables, such as the survival probability and the distribution of +the maximum position of the random walk that we study in this paper. In the next section, for clarity, we summarize +our main results. Then, in the following sections, we discuss each result in detail. +III. +SUMMARY OF KEY RESULTS +In this paper we derive exact results in the scaling limit for three observables: the position distribution, the survival +probability and the distribution of the maximum of the random walk. For clarity, we state these results here; their +derivations will be presented in the following sections. +Position distribution: in the large t and large k limit, such that z = k/(3t)1/3 is fixed, the position distribution of +the walker P(k, t) is given by +P(k, t) → +1 +(3t)1/3 G +� +k +(3t)1/3 +� +(8) +where the scaling function G(z) is given by +G(z) = +31/3 +2 Γ(2/3) |z| e−|z|3/3 . +(9) +This function has a trough at z = 0 and the function is bimodal with peaks at z = ±1 (see Fig. 2). +Survival probability: For a walker that starts from k0 > 0, the probability that the trap at k = 0 has not been +visited by time t is equivalent to the survival probability Q(k0, t) in the presence of an absorbing site at the +origin k = 0. This is given in the scaling limit by +Q(k0, t) ≈ f +� +k0 +(3t)1/3 +� +, +where +f(z) = 1 − +1 +Γ(1/3) Γ(1/3, z3/3) , +(10) +(see Fig. 3). This implies that in the long time limit the survival probability decays as t−1/3 (see equation (26)). +We also compute the joint distribution of position and survival (equations (31) and (34)). + +5 +Distribution of maximum: The distribution of M, the furthest site to the right visited by the walker up to time +t, or equivalently the deepest trap visited up to time t, is given in the scaling limit by +P(M = L, t) → +1 +(3t)1/3 g +� +L +(3t)1/3 +� +(11) +where y = L/t1/3 is now the scaling variable. The scaling function g(y) (see Fig. 5) is described in equations +(56),(58) and (59). +IV. +SCALING FORM OF THE POSITION DISTRIBUTION +−3 +−2 +−1 +0 +1 +2 +3 +k/(3t) +1 +3 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +(3t) +1 +3P(k) +Position Distribution +Scaling Function G(z) +FIG. 2. The scaling function G(z) plotted as a function of z. Symbols are obtained from Monte Carlo simulation data for the +random walk. Starting from k0 = 0 at t = 0, the random walk is numerically evolved up to t = 20000. The symbols show the +scaled histogram of final positions obtained from n = 150000 runs of the random walk simulation. +We now derive equations (8) and (9) for the position distribution P(k, t) of the random walk. As discussed earlier, +the form of equation (3) implies that the correct scaling variable involving k and t is z = k/(bt1/3), and we choose the +arbitrary constant as b = 31/3 for later convenience. Therefore, to solve the hydrodynamic equation (3), we assume a +scaling solution at late times and large k of the form +P(k, t) = +1 +(3t)1/3 G +� +k +(3t)1/3 +� +, +(12) +where G(z) is symmetric around z = 0, and is normalized to 1, i.e., +� ∞ +−∞ G(z) dz = 1, or equivalently +� ∞ +0 +G(z) dz = 1 +2 . +(13) + +6 +Substituting the scaling ansatz (12) in equation (1) and taking the scaling limit k → ∞, t → ∞ keeping z = k/(3t)1/3 +fixed, we find that G(z), for z > 0, satisfies a second order ordinary differential equation +G′′(z) + +� +z2 − 2 +z +� +G′(z) + +� +z + 2 +z2 +� +G(z) = 0 . +(14) +Remarkably, the general solution of this differential equation can be expressed in a simple closed form +G(z) = c1 z e−z3/3 + c2 z e−z3/3 +� z +0 +eu3/3 du , +(15) +where c1 and c2 are arbitrary. However, the second solution (the second term in (15)) behaves, for large z, as 1/z, +and hence is not normalisable, implying that we must have c2 = 0. The constant c1 can be fixed via the normalization +constant +� ∞ +0 +G(z)dz = 1/2. Using the symmetry around z = 0, the full solution for the scaling distribution (12) is +then given by +G(z) = +31/3 +2 Γ(2/3) |z| e−|z|3/3 . +(16) +This function is plotted in Fig. +(2) where we also plot results of Monte Carlo simulations that approach the +scaling curve. Strikingly, in contrast to a simple random walk (where the scaling variable is z = k/(2t)1/2 and the +corresponding scaling function is Gaussian with a peak at z = 0), G(z) has a trough at z = 0 where the solution has a +cusp singularity. The origin of this trough can be traced back to the drift term (away from the origin) in the current +in equation (5), that leads to a depletion of probability density near the origin at long times. Thus by creating an +emergent current away from the origin, the sluggish dynamics that is manifested in our model keeps the particle away +from the origin and produces two peaks (i.e. bimodality) in the probability distribution; these peaks are located at +z = ±1 or equivalently |k| = (3t)1/3. The distribution of the depth of the trap occupied at time t also follows from +equation (8) since the trap depth is a ln(|k| + 2). +V. +SURVIVAL PROBABILITY +We now introduce a sink at the origin, such that if the random walker arrives at k = 0, it dies. We consider the +survival probability of the walker in the presence of this absorbing site at k = 0. The ‘survival probability’ Q(k0, t) +denotes the probability that the walker is still alive after t steps, given that it starts at site k0 at time zero. Clearly, +Q(k0, t) is symmetric in k0, so we will consider only k0 ≥ 0, implying the walk that is defined on the positive integers. +It is convenient to use the backward master equation for the survival probability: +Q(k0, t + 1) = +1 +k0 + 2 Q(k0 + 1, t) + +1 +k0 + 2 Q(k0 − 1, t) + +� +1 − +2 +k0 + 2 +� +Q(k0, t) , +(17) +for k0 ≥ 1. This equation has a simple interpretation, corresponding to the events that may occur in the first step +of the walk. In the first step, the walker either hops from site k0 (rightwards to k0 + 1, or leftwards to k0 − 1), or it +stays at k0. Then, starting from its position at time step 1, it has to survive a further t steps. Summing these three +possibilities for the first step leads to equation (17), which needs to be solved for k0 ≥ 1 with the boundary conditions +Q(k0 = 0, t) = 0 +(18) +Q(k0 → ∞, t) = 1 . +(19) +The first boundary condition corresponds to the fact that if the walker starts at the absorbing site k0 = 0 it dies +immediately. The second condition follows from the fact that if the walker starts far away from the origin, it survives +with probability 1 as long as t is finite. In the limit of continuous time t and space k the backward equation becomes +∂ +∂tQ(k0, t) = 1 +k0 +∂2 +∂k2 +0 +Q(k0, t) . +(20) +It is convenient to use a scaling approach to quickly derive the large t asymptotic behaviour of the survival prob- +ability. We aim to solve equation (20) in the scaling limit introduced in section IV when both k0 and t are large. +Following the discussion in section IV we expect that Q(k0, t) will satisfy a scaling form +Q(k0, t) → f +� +k0 +(3t)1/3 +� +, +(21) + +7 +where f(z) is the scaling function. We now substitute the scaling form (21) in equation (20) and expand to leading +order to obtain the following second order ordinary differential equation in z ≥ 0 for the scaling function +f ′′(z) = −z2 f ′(z) , +(22) +subject to the two boundary conditions +f(z = 0) = 0 +and +f(z → ∞) = 1 , +(23) +which follow from Eqs. (18) and (19) respectively. +0 +2 +4 +6 +8 +10 +12 +14 +k0/(3t) +1 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Q(k0, t) +k0 = 25, t ∈ [1, 30000] +k0 = 50, t ∈ [1, 30000] +k0 = 100, t ∈ [1, 30000] +k0 = 400, t ∈ [1, 30000] +Scaling Function f(z) +FIG. 3. Full curve: the scaling function f(z) plotted as a function of scaling variable z. Symbols are obtained from Monte +Carlo simulation data for different values of k0. For a given k0, the random walk is numerically evolved over the time window +shown in the legend. The symbols show histograms obtained from n = 10000 runs of the random walk simulation. These +histograms are plotted against the scaling variable z = k0/(3t)1/3 . The different intervals of z for different values of k0 are +chosen for purposes of clarity. +The solution of equation (22) can be found trivially. Integrating (22) once gives f ′(z) = C e−z3/3. Integrating once +more, using the boundary conditions (23), leads to the exact solution for the scaling function: +f(z) = +� z +0 e−x3/3 dx +� ∞ +0 +e−x3/3 dx = 1 − +1 +Γ(1/3) Γ(1/3, z3/3) , +(24) +where Γ(s, x) = +� ∞ +x e−t ts−1 dt is the incomplete Gamma function. The scaling function f(z) is plotted in Fig. (3). +It is linear for small z (t1/3 ≫ k0) and saturates at f = 1 for large z (t1/3 ≪ k0). More precisely, the scaling function +has the asymptotic behaviours +f(z) ≈ +� +� +� +� +� +32/3 +Γ(1/3) z + O(z4) +as z → 0 +1 − +32/3 +Γ(1/3) z2 e−z3/3 +as z → ∞ . +(25) + +8 +In particular, for z → 0 +Q(k0, t) ≃ +31/3 +Γ(1/3) +k0 +t1/3 . +(26) +Equation (26) implies that in the limit t → ∞ the asymptotic behaviour of the survival probability is Q ∼ t−1/3. The +exponent 1/3 is smaller than the value 1/2 that is obtained for a simple diffusive process, implying that the decay +is slower than for simple diffusion. Again, the sluggish dynamics results in a significant difference in the dynamical +properties, compared to those of a simple random walk. +VI. +JOINT SURVIVAL AND POSITION DISTRIBUTION +Next we consider the probability Ps(k, t|k0) that a walker, starting at k0 > 0 at time t = 0, arrives at k at time t, +having in the meantime avoided the sink at k = 0. This is the joint distribution of survival and position, with the +subscript s in Ps(k, t|k0) denoting survival. +For this calculation we use the forward master equation, for k > 0: +Ps(k, t + 1|k0) = +1 +k + 3 Ps(k + 1, t|k0) + +1 +k + 1 Ps(k − 1, t|k0) + +k +k + 2 Ps(k, t|k0) , +(27) +with the boundary condition Ps(0, t|k0) = 0 and the initial condition, Ps(k, t = 0|k0) = δk,k0. When summed over +k = 1, 2, · · · , one should recover the survival probability of section V, namely +∞ +� +k=1 +Ps(k, t|k0) = Q(k0, t) . +(28) +-4 +-2 +2 +4 z +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +H(z) +FIG. 4. The scaling function H(z), given by equation (34), plotted as a function of z. +For simplicity, we will again work in the scaling limit where t → ∞, k → ∞ and k0 → ∞, keeping z = k/(3t)1/3 +and y = k0/(3t)1/3 fixed. We expect a scaling form +Ps(k, t|k0) ≈ +1 +(3t)1/3 W +� +k +(3t)1/3 , +k0 +(3t)1/3 +� +, +(29) +such that when integrated over k, we recover the scaling of the survival probability survival probability Q(k0, t) in +equation (24) with +� ∞ +0 +W(z, y) dz = f(y) , +(30) + +9 +where f(y) is given in equation (24). Here we assume k0 ∼ O(1), so that the second argument of the scaling function +W in equation (29) approaches zero. From the small argument behaviour of the survival probability in (26), we expect +that W(z, y → 0) → y H(z). This leads us to the scaling ansatz, valid for any k0 ∼ O(1): +Ps(k, t|k0) ≈ +k0 +(3t)2/3 H +� +k +(3t)1/3 +� +. +(31) +Substituting this scaling ansatz in equation (27), we get, to leading order in 1/t, the following ordinary differential +equation for H(z), for any z ≥ 0 (for z < 0, this function is symmetric, hence we consider only z ≥ 0): +H′′(z) + +� +z2 − 2 +z +� +H′(z) + +� +2z + 2 +z2 +� +H(z) = 0 . +(32) +The scaling function H(z) should satisfy the absorbing boundary condition H(0) = 0. One more condition can be +derived by substituting the scaling ansatz (31) in equation (28), and taking the limit y = k0/(3t)1/3 → 0. Using the +small y behaviour of f(y) in equation (25), we obtain the following condition: +� ∞ +0 +H(z) dz = +32/3 +Γ(1/3) . +(33) +One can easily check that the normalised solution of (32) is simply +H(z) = +32/3 +Γ(1/3) z2e−z3/3 . +(34) +H(z) is plotted in Fig. (4). We note that the trough around z = 0 is quadratic in z for this calculation in the presence +of a sink, in contrast to the linear |z| dependence for the trough in the position distribution for the calculation without +a sink (equation (16)). The quadratic behaviour of H(z) near the origin also contrasts with the analogous result for +the simple random walk case where linear behaviour is obtained as z → 0. This limit z → 0 gives information on the +long time behaviour; from (31),(34) we obtain +Ps(k, t|k0) ≃ +k0k2 +32/3Γ(1/3) +1 +t4/3 . +(35) +The t−4/3 long-time behaviour of the survival probability in equation (35) contrasts with the corresponding t−3/2 +behaviour for a simple random walk. +VII. +DISTRIBUTION OF THE MAXIMUM OF THE RANDOM WALK +We now remove the sink at the origin and instead consider a walker that starts at the origin (k0 = 0) and moves +freely. We study the statistics of its maximum displacement M(t) on the positive side up to time t. This corresponds +to the deepest trap visited to the right of the origin up to time t. Then the cumulative distribution Prob. [M(t) ≤ L] +is just the probability that the walker, starting at the origin, does not visit the site L up to time t. Let S(k0, t) denote +the probability that starting from k0 at t = 0, the walker does not visit L up to t. We then have +Prob. [M(t) ≤ L] = S(0, t) . +(36) +To compute S(0, t), we will first solve S(k0, t) for a general starting point k0 and then set k0 = 0. The survival +probability S(k0, t) again evolves according to the backward master equation +S(k0, t + 1) = +1 +|k0| + 2 S(k0 + 1, t) + +1 +|k0| + 2 S(k0 − 1, t) + +� +1 − +2 +|k0| + 2 +� +S(k0, t) , +(37) +with boundary condition +S(L, t) = 0 , +(38) +i.e. we impose a sink at site k = L. The initial condition (starting from k0 < L) is +S(k0, 0) = 1 . +(39) + +10 +Following the approach of section V, we expand in k0 to obtain the backward Fokker Planck equation: +∂ +∂tS(k0, t) = +1 +|k0| +∂2 +∂k2 +0 +S(k0, t) , +(40) +which is valid for k0 ≤ L, with an absorbing boundary condition S(k0 = L, t) = 0 at the sink k = L and the initial +condition S(k0, 0) = 1 for all k0 < L. +To solve equation (40), it is convenient to consider the Laplace transform +�S(k0, s) = +� ∞ +0 +S(k0, t) e−s t dt . +(41) +This satisfies +∂2 +∂k2 +0 +�S(k0, s) = s|k0|�S(k0, s) − |k0| , +(42) +where we used the initial condition S(k0, 0) = 1. Due to the presence of the absolute value k0 in the differential +equation (42), we need to solve for 0 ≤ k0 ≤ L and k0 ≤ 0 separately, and then match the solution and its first +derivative at k0 = 0. +The general solution of (42) for 0 ≤ k0 ≤ L and k0 ≤ 0 reads +�S(k0, s) = 1 +s + a1 Ai(s1/3k0) + b1 Bi(s1/3k0) +for +0 ≤ k0 ≤ L +(43) +�S(k0, s) = 1 +s + a2 Ai(−s1/3k0) +for +k0 ≤ 0 , +(44) +where Ai(x) and Bi(x) are the two linearly independent solutions of the Airy differential equation U ′′(x)−xU(x) = 0. +Since Bi(−x) diverges as x → −∞, we discarded this in the solution for k0 ≤ 0 in equation (44). The three constants +(independent of k0) a1, a2, b1 are fixed by the continuity of �S(k0, s), the continuity of ∂k0 �S(k0, s) at k0 = 0 and +the absorbing boundary condition �S(k0 = L, s) = 0, which yield three linear equations. These three constants can +then be straightforwardly determined explicitly (we do not give the details here). If the walker starts at k0 = 0 (for +simplicity), from equation (44), we just need the constant a2(s) since +�S(0, s) = 1 +s + a2(s) Ai(0) . +(45) +It turns out that the expression of a2(s) is rather simple: +a2(s) = +1 +2π Ai(0) Ai′(0) s Bi(s1/3 L) = − +√ +3 +s Bi(s1/3 L) , +(46) +where we used Ai(0) = 3−2/3/Γ(2/3) and Ai′(0) = −3−1/3/Γ(1/3). Plugging in equation (45) then gives the exact +Laplace transform, valid for all s: +�S(0, s) = 1 +s +� +1 − +1 +31/6 Γ(2/3) +1 +Bi(s1/3 L) +� +. +(47) +Taking the Laplace transform of equation (36), and plugging in the result (47), we obtain the exact Laplace +transform of the cumulative distribution of the maximum: +� ∞ +0 +Prob. [M(t) ≤ L] e−s t dt = 1 +s +� +1 − +1 +31/6 Γ(2/3) +1 +Bi(s1/3 L) +� +. +(48) +This result can be further simplified by noting that Prob. [M(t) ≥ L] = 1 − Prob. [M(t) ≤ L]. Consequently, +� ∞ +0 +Prob. [M(t) ≥ L] e−s t dt = +1 +31/6 Γ(2/3) +1 +s Bi(s1/3 L) . +(49) +Formally inverting this Laplace transform using the Bromwich contour and rescaling sL1/3 = λ, one sees immediately +that for all t and L, the cumulative distribution takes the scaling form +Prob. [M(t) ≥ L] = Y +� +t +L1/3 +� +, +(50) + +11 +10−1 +100 +m/(3t) +1 +3 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +(3t) +1 +3P(m) +Distribution of Maximum +Higher End Tail of g(z) +Lower End Tail of g(z) +FIG. 5. Distribution of the maximum of the random walk. The two full curves denote the lower end tail of g(z), equation (59), +and higher end tail of g(z), equation (58). The symbols are obtained from Monte Carlo simulation data for the random walk. +Starting from k0 = 0 at t = 0, the random walk was evolved up to t = 20000. The symbols show the scaled histogram obtained +from n = 105000 runs of the random walk simulation. +where the scaling function Y (y) has the exact Laplace transform +� ∞ +0 +e−λy Y (y) dy = 31/3Γ(1/3) +2π +1 +λBi(λ1/3) . +(51) +While it is difficult to invert the Laplace transform exactly, it is straightforward to extract its asymptotic behaviours, +as shown below. +The large y behaviour of Y (y) is controlled by the small λ expansion of (51) +� ∞ +0 +e−λyY (y) dy ≃ 1 +λ − 31/3Γ(2/3) +Γ(1/3) +1 +λ2/3 + 32/3Γ2(2/3) +Γ2(1/3) +1 +λ1/3 + . . . , +(52) +which yields the large y asymptotic expansion +Y (y) ∼ 1 − +31/3 +Γ(1/3) +1 +y1/3 + 32/3Γ2(2/3) +Γ3(1/3) +1 +y2/3 . . . . +(53) +The small y behaviour of Y (y) can be obtained from the large λ asymptotic behaviour of (51) +� ∞ +0 +e−λyY (y) dy ∼ +π1/2 +31/6Γ(2/3)λ11/12 e−2/3 λ1/2 , +(54) +which can be inverted to give the small y behaviour +Y (y) ≃ +32/3 +Γ(2/3)y1/3e−1/(9y) . +(55) + +12 +Using equation (50), we can now express the probability density Prob. [M(t) = L] of the maximum of the random +walk in a scaling form: +Prob. [M(t) = L] = − d +dLProb. [M(t) ≥ L] = +1 +(3t)1/3 g +� +L +(3t)1/3 +� +, +(56) +where the scaling function g(z) is simply related to the scaling function Y (y) and we deduce that +g(z) = z−4 Y ′(y)|y=1/(3z3) . +(57) +Using the asymptotic behavior of Y (y), we can then obtain the asymptotic tails of g(z) as +g(z) ∼ +31/3 +Γ(2/3) z e−z3/3 +for +z → ∞ +(58) +g(z) ∼ +32/3 +Γ(1/3) +� +1 − 2.32/3Γ2(2/3) +Γ2(1/3) +z . . . +� +for +z ≪ 1 . +(59) +In Fig. (5) the tails of g(z), equations (58) and (59), are compared with the numerical simulations. It is useful to +compare equation (58) with equation (16). We see that for large z, the scaling function of the position distribution +(16) and that of the maximum (58) have the same asymptotic tails up to an overall factor 1/2. This is similar to +what occurs for a simple random walk, although in that case the tails are Gaussian. The small z behaviour (59) for +the scaling function is a constant with a linear correction. The constant is consistent with the large time limit of the +survival probability (26). The linear correction contrasts with the case of a simple random walk where the correction +to the constant term is quadratic in the scaling variable. +VIII. +GENERATING FUNCTION APPROACH +In sections IV - VII, we adopted a scaling approach to obtain long-time asymptotic results for the sluggish random +walk problem. We now illustrate how a generating function approach may be employed to find the exact solution +for all times. We will see that the long time limit of the solution obtained using the generating function approach +recovers the results of the scaling approach. For the sake of brevity, we restrict ourselves to the computation of the +survival probability. +Consider again Q(k0, t), the survival probability for a walker starting at k0 in the presence of a sink at the origin +k = 0. Q(k0, t) satisfies the backward master equation (17). We define a generating function with parameter λ: +G(k0) = +∞ +� +t=0 +λtQ(k0, t) . +(60) +Substituting (60) into (17) and imposing the initial condition Q(k0, 0) = 1, we obtain +� +k0 +1 − λ +λ ++ 2 +λ +� +G(k0) − G(k0 + 1) − G(k0 − 1) = k0 + 2 +λ +. +(61) +Equation (61), in which k0 takes integer values, can be solved using Bessel functions. However it is easier to take a +continuum limit and expand to second order in k0, to obtain +∂2G(k0) +∂k2 +0 +− +�(k0 + 2)(1 − λ) +λ +� +G(k0) = −(k0 + 2) +λ +. +(62) +The homogeneous version of (62) (i.e. equating the lhs to zero) has Airy functions as solutions: +Ghom(k0) = B0Ai(C(k0 + 2)) + B1Bi(C(k0 + 2)) , +(63) +where +C = +�1 − λ +λ +�1/3 +, +(64) + +13 +and the constants B0 and B1 are to be fixed by the boundary conditions. We can set B1 = 0 and discard the Bi +solution, as it diverges as k0 → ∞. +Then for G(k0) a particular solution to (62) is 1/(1 − λ) and the general solution to (62) is +G(k0) = B0Ai((k0 + 2)C) + +1 +1 − λ . +(65) +The boundary condition is G(0) = 0, which fixes the constant B0, and we obtain the solution to (62) as +G(k0) = +1 +1 − λ +� +1 − Ai((k0 + 2)C) +Ai(2C) +� +. +(66) +We are interested in the long time asymptotic behaviour, which we can extract from the λ → 1 limit of (66). Since +C → 0 as λ → 1, we require the small argument expansion of the Airy function: +Ai(x) ≃ +1 +32/3Γ(2/3) − +x +31/3Γ(1/3) . +(67) +We then find, in the limit λ → 1, +G(k0) ≃ 31/3 Γ(2/3) +Γ(1/3) +k0 +(1 − λ)2/3 . +(68) +Thus the leading singularity is at λ∗ = 1 and is of the form (λ∗ − λ)−2/3. Invoking the usual Tauberian theorem [33], +this singularity gives the following large t asymptotic behaviour: +Qt(k0) ∼ +31/3 +Γ(1/3)k0t−1/3 . +(69) +This matches perfectly with the small z asymptotic of the scaling behaviour in (24) upon using the small z expansion +of f(z) in equation (25). +IX. +GENERALISATION TO THE CASE WHERE α ̸= 1 +Up to now, we have considered only the case where the probability of hopping to the right or left is proportional +to 1/(|k| + 2), i.e. the exponent α = 1 in the general expression for the hopping probability, A(|k| + 2)−α. We now +generalise to the case where α ̸= 1, i.e. the hopping probability is proportional to 1/(|k| + 2)α. In this case equation +(3) generalises for k ≥ 0 to +∂ +∂tP(k, t) ≈ 1 +kα +� ∂2 +∂k2 P(k, t) − 2α +k +∂ +∂k P(k, t) + α(α + 1) +k2 +P(k, t) +� +. +(70) +Equation (70) can be put into the standard form (6), where now D(k) = 1/kα and U(k) = 1/kα. One can again +solve (70) by the scaling approach discussed earlier. For general positive α it is easy to show that the scaling variable +becomes k/tν where ν = (2 + α)−1. Therefore the solution of (70) for P(k, t) has a scaling form +P(k, t) = t− +1 +α+2 G +� +k t− +1 +α+2 +� +, +(71) +where the scaling function G(z) is symmetric and, for positive z, satisfies the nontrivial differential equation +G′′(z) + +� zα+1 +α + 2 − 2α +z +� +G′(z) + +� zα +α + 2 + α(α + 1) +z2 +� +G(z) = 0 , +(72) +with boundary condition G(z) → 0 as z → ∞. Remarkably, this equation admits the simple solution, satisfying the +boundary condition, +G(z) = A zα exp +� +− +zα+2 +(α + 2)2 +� +, +(73) + +14 +where the normalisation constant A is given by +A−1 = 2(α + 2) +α +α+2 Γ +�α + 1 +α + 2 +� +. +(74) +Using the symmetry G(z) = G(−z), the full solution for all z can be written as +G(z) = A |z|α exp +� +− |z|α+2 +(α + 2)2 +� +. +(75) +When α = 0 we recover the standard Gaussian result for a simple random walk, while for α = 1 we recover the result +(16) upon rescaling z → 31/3z. We note that for any α > 0 there is a trough, i.e. a cusp singularity, at z = 0. The +trough at z = 0 disappears only for the case of simple diffusion (α = 0). +Similar scaling analyses can be performed for the survival probability as well as the distribution of the maximum +site visited to the right. We do not repeat the analysis, but just note that the scaling implies that the asymptotic +decay of the survival probability is Q(t) ∼ t−1/(α+2) and the maximum scales as M(t) ∼ t1/(α+2) . +X. +CONCLUSION +In this paper we have studied a random walk with space-dependent transition probabilities. Our study was motivated +by trap models of slow dynamics, but in contrast to most such models, our trap depths are not random but instead +increase logarithmically with distance k from the origin. The dynamics of a particle moving on the lattice of traps +follows an inhomogeneous random walk which has symmetric transition probabilities that decrease with k as 1/k. Thus +the motion of a walker slows down as it goes further and further away from the origin, a phenomenon that we term +‘sluggish dynamics’. The sluggish dynamics causes the typical distance explored up to time t to grow subdiffusively +as t1/3, in contrast to the standard t1/2 law for a simple random walk. +We used a scaling approach, in which the scaling variable is k/t1/3, to compute long-time asymptotic results for +various properties of this inhomogeneous random walk: the position distribution, the survival probability in the +presence of a sink at the origin, the joint survival and position distribution, and the distribution of the maximum +distance to the right. Interestingly, the position distribution has a trough (a cusp singularity) at the origin and is +bimodal, with two peaks located at |k| = (3t)1/3. The contrasts with the usual Gaussian distribution for diffusion +(which has a single maximum at k = 0). The bimodal distribution and the t1/3 scaling reflect the sluggish nature of +the dynamics. The survival probability shows an asymptotic decay ∼ t−1/3 at large time, which contrasts with the +t−1/2 decay for a simple random walk. The fact that the survival probability decays to zero as t → ∞ implies that +the walk is recurrent in d = 1, as is the simple random walk. The distribution of the maximum of the walk up to time +t has a nontrivial scaling function. +We further showed how a generating function approach can be used to find exact solutions for all times. Using +this approach to compute the survival probability in the presence of a sink at the origin, we recover our scaling +result in the long-time limit. Application of the same generating function approach to other observables should be a +straightforward extension. +Finally, we generalised the model to cases where the transition probability decays as 1/|k|α with positive α. Except +for α = 0 (simple random walk), the position distribution always shows a trough at the origin (k = 0), where it +exhibits a singularity, behaving as |k|α. Remarkably, the scaling function for the position distribution takes on a +simple form (equation (75)) and there is always a trough at the origin with associated singularity |z|α for α > 0. +It is worthwhile comparing the behaviour of our sluggish random walk model with that of the Gillis model outlined +in the introduction. In the continuum limit the Gillis model becomes diffusion in a logarithmic potential [19, 24] and +the corresponding Fokker-Planck equation reads +∂ +∂tP(k, t) = ∂ +∂k +� ∂ +∂k P(k, t) + +� ∂ +∂k U(k) +� +P(k, t) +� +, +(76) +where the potential U(k) = 2ϵ ln |k|. The relevant case for us is ϵ < 0 whereby the potential is repulsive and the +particle is pushed away from the origin. In this Gillis case, the solution for the time-dependent position distribution +has scaling form [19, 24] +P(k, t) → +1 +t1/2 GGill +� k +t1/2 +� +(77) + +15 +where the scaling function, GGill(z), is given by +GGill(z) = +2ϵ−1/2 +Γ(1/2 − ϵ) |z|−2ϵ e−|z|2/2 . +(78) +This is to be compared with the scaling function G(z) (16) for the sluggish random walk model (where the scaling +variable is z = k/(3t)1/3). As with (16), the scaling function (78) is bimodal, with peaks at z = ±(−2ϵ)1/2, and has a +trough at the origin. However, the model exhibits diffusive scaling and is thus not sluggish. The difference between +the sluggish random walk and diffusion in a logarithmic potential is evident when one compares the Fokker Planck +equations (6) and (76). The key difference is the space-dependent diffusion constant D(k) = 1/k appearing in (76), +along with the potential U(k) = 1/k . It is these features that lead to a change of the scaling variable to z = k/(3t)1/3 +and consequent sluggish behaviour. +The sluggish random walk model and its analysis are straightforward to generalise to higher dimensions and other +observables. For example, it would interesting to study the return probabilities and recurrence/transience transition +in a higher dimension for general α. It would also be of interest to study the time for the walker to traverse from +one maximum of the position distribution to the other. More generally our study has shown that inhomogenous +space-dependent random walks can exhibit surprising properties and it remains to explore the full range of such +behaviour. +AZ acknowledges support of the INSPIRE fellowship from DST India and the Physics Computing Facility lab at +UCSD. RJA was supported by the European Research Council under consolidator grant 682237 EVOSTRUC and +by the Excellence Cluster Balance of the Microverse (EXC 2051 - Project-ID 390713860) funded by the Deutsche +Forschungsgemeinschaft (DFG). For the purpose of open access, the author has applied a Creative Commons Attribu- +tion (CC BY) licence to any Author Accepted Manuscript version arising from this submission. MRE thanks LPTMS +for the award of a CNRS Visiting Professorship, during which this work was written up. +[1] Wolynes P G, Lubchenko V (Editors) 2012 Structural Glasses and Supercooled Liquids: Theory, Experiment, and Appli- +cations. John Wiley & sons. +[2] Berthier L, Biroli G, Bouchaud J-P, Cipelletti,L, van Saarloos W. (Editors) 2011. Dynamical heterogeneities in glasses, +colloids, and granular media (Vol. 150). OUP Oxford. +[3] Bouchaud J-P 1992 Weak ergodicity breaking and aging in disordered systems J. Phys. 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CRC press. + diff --git a/-NFPT4oBgHgl3EQfZTSq/content/tmp_files/load_file.txt b/-NFPT4oBgHgl3EQfZTSq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..59f352a7c8057fad0f396550c2069f00a6e1ff98 --- /dev/null +++ b/-NFPT4oBgHgl3EQfZTSq/content/tmp_files/load_file.txt @@ -0,0 +1,475 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf,len=474 +page_content='A sluggish random walk with subdiffusive spread Aniket Zodage1,2, Rosalind J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Allen3,4, Martin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Evans4,5, Satya N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Majumdar5 1 Department of Physics, Indian Institute of Science Education and Research, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Homi Bhabha Road, Pune 411008, India 2 Department of Physics, UC San Diego, 9500 Gilman Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' La Jolla, California 92093, USA 3 Theoretical Microbial Ecology, Institute of Microbiology, Faculty of Biological Sciences, Friedrich Schiller University Jena, Buchaer Strasse 6, 07745 Jena, Germany 4 SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, EH9 3FD and 5 LPTMS, CNRS, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France (Dated: January 31, 2023) We study a one-dimensional sluggish random walk with space-dependent transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases logarithmically with distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This leads to a random walk which has symmetric transition probabilities that decrease with distance |k| from the origin as 1/|k| for large |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We show that the typical position after time t scales as t1/3 with a nontrivial scaling function for the position distribution which has a trough (a cusp singularity) at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Therefore biased random motion emerges even though the transition probabilities are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We also compute the survival probability of the walker in the presence of a sink at the origin and show that it decays as t−1/3 at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Furthermore we compute the distribution of the maximum position, M(t), to the right of the origin up to time t, and show that it has a nontrivial scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Finally we provide a generalisation of this model where the transition probabilities decay as 1/|k|α with α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='13077v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='stat-mech] 30 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' INTRODUCTION Slow dynamics is a common feature of many physical systems, including glasses, granular media and colloids [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' well Slow dynamics commonly arises when the system becomes trapped for increasing periods of time in deeper and deeper local free energy minima in the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This phenomenon has inspired the study of simplified toy models known as trap models [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In these models, the many minima of the complex disordered landscape are represented by traps whose depths are taken to be random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For a single particle hopping between nearby traps the mean squared displacement typically grows more slowly than linearly in time, thus the particle’s motion is subdiffusive [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Similar slow dynamics can also arise in an inhomogeneous landscape where the trap depth is position-dependent (as opposed to being random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Here, the hopping dynamics between the traps is an example of a Markov chain with space-dependent transition probabilities [11, 12], or in other words, an inhomogeneous random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For such systems, explicit solutions for observables beyond the simple position distribution – such as first passage probabilities [13–15] or extreme value statistics [16, 17] – are generally hard to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' A classic example of an inhomogenous random walk is the centrally-biased Gillis model [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In this model, a single particle hops on a one-dimensional lattice where the hopping probability is asymmetric in a position-dependent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Specifically, for k ̸= 0, the hopping probabilities from site k to k ± 1 are 1 2(1 ∓ ϵ/k), while for k = 0 the hopping probabilities to sites ±1 are both 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Because the hopping probabilities (for k ̸= 0) are asymmetric, the particle undergoes a biased random walk in which the parameter ϵ ∈ [−1, 1] controls the strength of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For ϵ > 0 there is a drift towards the origin while for ϵ < 0 there is a drift away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Far away from the origin, where |k| ≫ 1, the bias is small and the dynamics tends towards a symmetric random walk which, in the continuum limit, reduces to a particle moving in a logarithmic potential U(k) → 2ϵ ln |k| [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The Gillis model [18–22], and its continuous limit of particle motion in a logarithmic potential [19, 23–27], have aroused much interest because of their relevance to vortex dynamics, interactions between tracer particles in a driven fluid, cold atoms trapped in optical lattices and the nonequilibrium behaviour of systems with long-range interactions [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' These models have the appealing feature of allowing the exact calculation of various observables going beyond the position distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Motivated by these works, here we consider the counterpart problem of an inhomogeneous trap model in which the trap depth increases logarithmically with increasing distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The dynamics of a particle in this model corresponds to a symmetric random walk with space-dependent hopping probabilities that decrease inversely with distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This random walk has the interesting property of being ‘sluggish’ since the particle’s motion slows down as it goes further away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We show that the physics of this model is quite different from the previously studied case of a particle in a logarithmic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Instead, in the continuum limit and for |k| ≫ 1, our model corresponds to a particle moving in a potential U(k) ∼ 1/|k| with, additionally, a space-dependent diffusion constant that also decays as 1/|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The interplay of these two features leads to an emergent bias in the dynamics away from the origin, even though the hopping probabilities are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The position distribution has a non-trivial and non- Gaussian form in which distance scales with time as t1/3 at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Moreover, we show that other observables such as the survival probability in the presence of an absorbing site and the distribution of the maximum displacement to one side of the origin can be computed explicitly and exhibit non-trivial scaling behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Finally we discuss how the model can be easily generalised to higher dimensions and other space-dependent hopping probabilities, such as a 1/|k|α decay with exponent α > 0, without losing its solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' MODEL DEFINITION AND HYDRODYNAMIC LIMIT As discussed, we consider an ordered array of traps arranged on a one-dimensional lattice such that the depth of the trap at site k is a ln(|k| + 2) (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The corresponding Arrhenius escape rate from the trap at site k is A(|k| + 2)−α with α = β a, where A is an overall constant and β is the inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Without loss of generality we will set A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We will mostly focus on the case where α = 1, although we briefly discuss α ̸= 1 in Section IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We consider the discrete-time dynamics of a particle moving at random on this infinite one-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The key feature of the dynamics is that, as time progresses, the particle explores sites further and further away from the origin in which it gets trapped to a greater and greater extent because of the increasing trap depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Hence the particle is subject to diminishing transition probability for exiting the traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' At each integer time step t the particle’s position evolves according to the following rules (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' From a site k at time t, the particle hops to site k + 1 with probability 1/(|k| + 2), it hops to site k − 1 with equal probability 1/(|k|+2), or it stays at site k with the complementary probability |k|/(|k|+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The time is then updated to t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We note that (in contrast to the Gillis model [18–22]) the hopping probabilities are symmetric for all k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Schematic illustration of our model, showing the ordered arrangement of traps and the hopping probabilities 1/(|k|+2) for a particle to move to a nearest neighbour of site k on the lattice, as well as the probability |k|/(|k| + 2) of remaining at site k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' however they differ from those of a simple random walk everywhere except at the origin, k = 0, where the hopping probability is 1/2 to either of k = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Let P(k, t) denote the position distribution at time t for a particle that starts from k0 = 0 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The distribution evolves via the forward master equation: P(k, t + 1) = 1 |k + 1| + 2 P(k + 1, t) + 1 |k − 1| + 2 P(k − 1, t) + |k| |k| + 2 P(k, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (1) The initial condition is P(k, 0) = δk,0 and the boundary condition is P(k, t) → 0 as |k| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The solution P(k, t) is symmetric around k = 0, hence we can just focus on k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We first write (1) in in the more suggestive form P(k, t + 1) − P(k, t) = 1 |k + 1| + 2 P(k + 1, t) + 1 |k − 1| + 2 P(k − 1, t) − 2 |k| + 2 P(k, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (2) For large k > 0 and large t, we can expand the right hand side (rhs) of Equation (2) as a Taylor series in k and replace the left hand side (lhs) by a time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This gives, keeping all terms of the same order, the hydrodynamic equation that captures the behaviour of the system at long distance and at late time: ∂ ∂tP(k, t) ≈ 1 k � ∂2 ∂k2 P(k, t) − 2 k ∂ ∂k P(k, t) + 2 k2 P(k, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (3) This hydrodynamic equation can be written as a continuity equation: ∂ ∂tP(k, t) = − ∂ ∂k j(k, t) , (4) where the space-time dependent current density j(k, t) reads j(k, t) = −1 k ∂ ∂k P(k, t) + 1 k2 P(k, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (5) We identify the first term on the rhs of (5) as a diffusive probability current and the second term as a drift away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The original equation (3) can also be expressed in the standard Fokker-Planck form ∂ ∂tP(k, t) = ∂ ∂k � D(k) ∂ ∂k P(k, t) + � ∂ ∂k U(k) � P(k, t) � , (6) where D(k) = 1/k and U(k) = 1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Equation (6) shows clearly the two features of the dynamics that lead to non- trivial behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Firstly, the effective diffusion constant D(k) = 1/k is space dependent, such that it slows down 4 the dynamics as the particle moves away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Additionally, an effective external potential U(k) = 1/k emerges, which is repulsive, such that it pushes the particle away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This repulsion arises from the microscopic dynamics: the hopping probability from site k to k + 1 is 1/(k + 2), while the reverse event (from (k + 1) to k) has probability 1/(k + 3) < 1/(k + 2) for any k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Thus, even though the hopping probabilities out of site k (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' to (k + 1) or (k − 1)) are symmetric, the space-dependence of the hopping probability produces an outward bias away from the origin (symmetrically for k < 0) which leads to the drift term in the current in equation (5) in the hydrodynamic description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For large time and space we expect to see a scaling regime in which the probability distribution becomes a function of a combination k/tν (with ν > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The following argument suggests that ν takes the value 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Let us assume that after time t, the typical value of the position k is ktyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The number of steps N that have been taken by the particle will scale as N ∼ t/ktyp, since the time for one step is the typical escape time 1/ktyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Since all steps are equal in distance and the hopping probability is symmetric, the position scales with the number of steps in the same way as for a simple random walk, ktyp ∼ N 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Putting these arguments together we obtain ktyp ∼ N 1/2 ∼ (t/ktyp)1/2 which implies that the position of the particle scales with time as ktyp ∼ t1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' hence ν = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This scaling can be confirmed by assuming the following scaling form for the probability distribution P(k, t) in the limit when both k and t are large, keeping the ratio k/tν (with ν > 0) fixed: P(k, t) → 1 b tν G � k b tν � , (7) where G(z) is the scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We have also incorporated an adjustable constant b which can be chosen appro- priately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Substituting the scaling form (7) in equation (1), one readily finds that for leading order terms to be of the same order we must have ν = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For convenience we will also choose b = 31/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We will discuss the precise form of the scaling function G(z) in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The scaling k ∼ t1/3 also manifests itself in other observables, such as the survival probability and the distribution of the maximum position of the random walk that we study in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In the next section, for clarity, we summarize our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Then, in the following sections, we discuss each result in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' SUMMARY OF KEY RESULTS In this paper we derive exact results in the scaling limit for three observables: the position distribution, the survival probability and the distribution of the maximum of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For clarity, we state these results here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' their derivations will be presented in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Position distribution: in the large t and large k limit, such that z = k/(3t)1/3 is fixed, the position distribution of the walker P(k, t) is given by P(k, t) → 1 (3t)1/3 G � k (3t)1/3 � (8) where the scaling function G(z) is given by G(z) = 31/3 2 Γ(2/3) |z| e−|z|3/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (9) This function has a trough at z = 0 and the function is bimodal with peaks at z = ±1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Survival probability: For a walker that starts from k0 > 0, the probability that the trap at k = 0 has not been visited by time t is equivalent to the survival probability Q(k0, t) in the presence of an absorbing site at the origin k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This is given in the scaling limit by Q(k0, t) ≈ f � k0 (3t)1/3 � , where f(z) = 1 − 1 Γ(1/3) Γ(1/3, z3/3) , (10) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This implies that in the long time limit the survival probability decays as t−1/3 (see equation (26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We also compute the joint distribution of position and survival (equations (31) and (34)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 5 Distribution of maximum: The distribution of M, the furthest site to the right visited by the walker up to time t, or equivalently the deepest trap visited up to time t, is given in the scaling limit by P(M = L, t) → 1 (3t)1/3 g � L (3t)1/3 � (11) where y = L/t1/3 is now the scaling variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The scaling function g(y) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 5) is described in equations (56),(58) and (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' SCALING FORM OF THE POSITION DISTRIBUTION −3 −2 −1 0 1 2 3 k/(3t) 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='40 (3t) 1 3P(k) Position Distribution Scaling Function G(z) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The scaling function G(z) plotted as a function of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Symbols are obtained from Monte Carlo simulation data for the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Starting from k0 = 0 at t = 0, the random walk is numerically evolved up to t = 20000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The symbols show the scaled histogram of final positions obtained from n = 150000 runs of the random walk simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We now derive equations (8) and (9) for the position distribution P(k, t) of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' As discussed earlier, the form of equation (3) implies that the correct scaling variable involving k and t is z = k/(bt1/3), and we choose the arbitrary constant as b = 31/3 for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Therefore, to solve the hydrodynamic equation (3), we assume a scaling solution at late times and large k of the form P(k, t) = 1 (3t)1/3 G � k (3t)1/3 � , (12) where G(z) is symmetric around z = 0, and is normalized to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=', � ∞ −∞ G(z) dz = 1, or equivalently � ∞ 0 G(z) dz = 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (13) 6 Substituting the scaling ansatz (12) in equation (1) and taking the scaling limit k → ∞, t → ∞ keeping z = k/(3t)1/3 fixed, we find that G(z), for z > 0, satisfies a second order ordinary differential equation G′′(z) + � z2 − 2 z � G′(z) + � z + 2 z2 � G(z) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (14) Remarkably, the general solution of this differential equation can be expressed in a simple closed form G(z) = c1 z e−z3/3 + c2 z e−z3/3 � z 0 eu3/3 du , (15) where c1 and c2 are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' However, the second solution (the second term in (15)) behaves, for large z, as 1/z, and hence is not normalisable, implying that we must have c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The constant c1 can be fixed via the normalization constant � ∞ 0 G(z)dz = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Using the symmetry around z = 0, the full solution for the scaling distribution (12) is then given by G(z) = 31/3 2 Γ(2/3) |z| e−|z|3/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (16) This function is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (2) where we also plot results of Monte Carlo simulations that approach the scaling curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Strikingly, in contrast to a simple random walk (where the scaling variable is z = k/(2t)1/2 and the corresponding scaling function is Gaussian with a peak at z = 0), G(z) has a trough at z = 0 where the solution has a cusp singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The origin of this trough can be traced back to the drift term (away from the origin) in the current in equation (5), that leads to a depletion of probability density near the origin at long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Thus by creating an emergent current away from the origin, the sluggish dynamics that is manifested in our model keeps the particle away from the origin and produces two peaks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' bimodality) in the probability distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' these peaks are located at z = ±1 or equivalently |k| = (3t)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The distribution of the depth of the trap occupied at time t also follows from equation (8) since the trap depth is a ln(|k| + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' SURVIVAL PROBABILITY We now introduce a sink at the origin, such that if the random walker arrives at k = 0, it dies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We consider the survival probability of the walker in the presence of this absorbing site at k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The ‘survival probability’ Q(k0, t) denotes the probability that the walker is still alive after t steps, given that it starts at site k0 at time zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Clearly, Q(k0, t) is symmetric in k0, so we will consider only k0 ≥ 0, implying the walk that is defined on the positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' It is convenient to use the backward master equation for the survival probability: Q(k0, t + 1) = 1 k0 + 2 Q(k0 + 1, t) + 1 k0 + 2 Q(k0 − 1, t) + � 1 − 2 k0 + 2 � Q(k0, t) , (17) for k0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This equation has a simple interpretation, corresponding to the events that may occur in the first step of the walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In the first step, the walker either hops from site k0 (rightwards to k0 + 1, or leftwards to k0 − 1), or it stays at k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Then, starting from its position at time step 1, it has to survive a further t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Summing these three possibilities for the first step leads to equation (17), which needs to be solved for k0 ≥ 1 with the boundary conditions Q(k0 = 0, t) = 0 (18) Q(k0 → ∞, t) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (19) The first boundary condition corresponds to the fact that if the walker starts at the absorbing site k0 = 0 it dies immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The second condition follows from the fact that if the walker starts far away from the origin, it survives with probability 1 as long as t is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In the limit of continuous time t and space k the backward equation becomes ∂ ∂tQ(k0, t) = 1 k0 ∂2 ∂k2 0 Q(k0, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (20) It is convenient to use a scaling approach to quickly derive the large t asymptotic behaviour of the survival prob- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We aim to solve equation (20) in the scaling limit introduced in section IV when both k0 and t are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Following the discussion in section IV we expect that Q(k0, t) will satisfy a scaling form Q(k0, t) → f � k0 (3t)1/3 � , (21) 7 where f(z) is the scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We now substitute the scaling form (21) in equation (20) and expand to leading order to obtain the following second order ordinary differential equation in z ≥ 0 for the scaling function f ′′(z) = −z2 f ′(z) , (22) subject to the two boundary conditions f(z = 0) = 0 and f(z → ∞) = 1 , (23) which follow from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (18) and (19) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 k0/(3t) 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='0 Q(k0, t) k0 = 25, t ∈ [1, 30000] k0 = 50, t ∈ [1, 30000] k0 = 100, t ∈ [1, 30000] k0 = 400, t ∈ [1, 30000] Scaling Function f(z) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Full curve: the scaling function f(z) plotted as a function of scaling variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Symbols are obtained from Monte Carlo simulation data for different values of k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For a given k0, the random walk is numerically evolved over the time window shown in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The symbols show histograms obtained from n = 10000 runs of the random walk simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' These histograms are plotted against the scaling variable z = k0/(3t)1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The different intervals of z for different values of k0 are chosen for purposes of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The solution of equation (22) can be found trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Integrating (22) once gives f ′(z) = C e−z3/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Integrating once more, using the boundary conditions (23), leads to the exact solution for the scaling function: f(z) = � z 0 e−x3/3 dx � ∞ 0 e−x3/3 dx = 1 − 1 Γ(1/3) Γ(1/3, z3/3) , (24) where Γ(s, x) = � ∞ x e−t ts−1 dt is the incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The scaling function f(z) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' It is linear for small z (t1/3 ≫ k0) and saturates at f = 1 for large z (t1/3 ≪ k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' More precisely, the scaling function has the asymptotic behaviours f(z) ≈ � � � � � 32/3 Γ(1/3) z + O(z4) as z → 0 1 − 32/3 Γ(1/3) z2 e−z3/3 as z → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (25) 8 In particular, for z → 0 Q(k0, t) ≃ 31/3 Γ(1/3) k0 t1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (26) Equation (26) implies that in the limit t → ∞ the asymptotic behaviour of the survival probability is Q ∼ t−1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The exponent 1/3 is smaller than the value 1/2 that is obtained for a simple diffusive process, implying that the decay is slower than for simple diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Again, the sluggish dynamics results in a significant difference in the dynamical properties, compared to those of a simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' JOINT SURVIVAL AND POSITION DISTRIBUTION Next we consider the probability Ps(k, t|k0) that a walker, starting at k0 > 0 at time t = 0, arrives at k at time t, having in the meantime avoided the sink at k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This is the joint distribution of survival and position, with the subscript s in Ps(k, t|k0) denoting survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For this calculation we use the forward master equation, for k > 0: Ps(k, t + 1|k0) = 1 k + 3 Ps(k + 1, t|k0) + 1 k + 1 Ps(k − 1, t|k0) + k k + 2 Ps(k, t|k0) , (27) with the boundary condition Ps(0, t|k0) = 0 and the initial condition, Ps(k, t = 0|k0) = δk,k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' When summed over k = 1, 2, · · · , one should recover the survival probability of section V, namely ∞ � k=1 Ps(k, t|k0) = Q(k0, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (28) 4 2 2 4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='6 H(z) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The scaling function H(z), given by equation (34), plotted as a function of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For simplicity, we will again work in the scaling limit where t → ∞, k → ∞ and k0 → ∞, keeping z = k/(3t)1/3 and y = k0/(3t)1/3 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We expect a scaling form Ps(k, t|k0) ≈ 1 (3t)1/3 W � k (3t)1/3 , k0 (3t)1/3 � , (29) such that when integrated over k, we recover the scaling of the survival probability survival probability Q(k0, t) in equation (24) with � ∞ 0 W(z, y) dz = f(y) , (30) 9 where f(y) is given in equation (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Here we assume k0 ∼ O(1), so that the second argument of the scaling function W in equation (29) approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' From the small argument behaviour of the survival probability in (26), we expect that W(z, y → 0) → y H(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This leads us to the scaling ansatz, valid for any k0 ∼ O(1): Ps(k, t|k0) ≈ k0 (3t)2/3 H � k (3t)1/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (31) Substituting this scaling ansatz in equation (27), we get, to leading order in 1/t, the following ordinary differential equation for H(z), for any z ≥ 0 (for z < 0, this function is symmetric, hence we consider only z ≥ 0): H′′(z) + � z2 − 2 z � H′(z) + � 2z + 2 z2 � H(z) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (32) The scaling function H(z) should satisfy the absorbing boundary condition H(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' One more condition can be derived by substituting the scaling ansatz (31) in equation (28), and taking the limit y = k0/(3t)1/3 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Using the small y behaviour of f(y) in equation (25), we obtain the following condition: � ∞ 0 H(z) dz = 32/3 Γ(1/3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (33) One can easily check that the normalised solution of (32) is simply H(z) = 32/3 Γ(1/3) z2e−z3/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (34) H(z) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We note that the trough around z = 0 is quadratic in z for this calculation in the presence of a sink, in contrast to the linear |z| dependence for the trough in the position distribution for the calculation without a sink (equation (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The quadratic behaviour of H(z) near the origin also contrasts with the analogous result for the simple random walk case where linear behaviour is obtained as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This limit z → 0 gives information on the long time behaviour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' from (31),(34) we obtain Ps(k, t|k0) ≃ k0k2 32/3Γ(1/3) 1 t4/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (35) The t−4/3 long-time behaviour of the survival probability in equation (35) contrasts with the corresponding t−3/2 behaviour for a simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' DISTRIBUTION OF THE MAXIMUM OF THE RANDOM WALK We now remove the sink at the origin and instead consider a walker that starts at the origin (k0 = 0) and moves freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We study the statistics of its maximum displacement M(t) on the positive side up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This corresponds to the deepest trap visited to the right of the origin up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Then the cumulative distribution Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≤ L] is just the probability that the walker, starting at the origin, does not visit the site L up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Let S(k0, t) denote the probability that starting from k0 at t = 0, the walker does not visit L up to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We then have Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≤ L] = S(0, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (36) To compute S(0, t), we will first solve S(k0, t) for a general starting point k0 and then set k0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The survival probability S(k0, t) again evolves according to the backward master equation S(k0, t + 1) = 1 |k0| + 2 S(k0 + 1, t) + 1 |k0| + 2 S(k0 − 1, t) + � 1 − 2 |k0| + 2 � S(k0, t) , (37) with boundary condition S(L, t) = 0 , (38) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' we impose a sink at site k = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The initial condition (starting from k0 < L) is S(k0, 0) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (39) 10 Following the approach of section V, we expand in k0 to obtain the backward Fokker Planck equation: ∂ ∂tS(k0, t) = 1 |k0| ∂2 ∂k2 0 S(k0, t) , (40) which is valid for k0 ≤ L, with an absorbing boundary condition S(k0 = L, t) = 0 at the sink k = L and the initial condition S(k0, 0) = 1 for all k0 < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' To solve equation (40), it is convenient to consider the Laplace transform �S(k0, s) = � ∞ 0 S(k0, t) e−s t dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (41) This satisfies ∂2 ∂k2 0 �S(k0, s) = s|k0|�S(k0, s) − |k0| , (42) where we used the initial condition S(k0, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Due to the presence of the absolute value k0 in the differential equation (42), we need to solve for 0 ≤ k0 ≤ L and k0 ≤ 0 separately, and then match the solution and its first derivative at k0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The general solution of (42) for 0 ≤ k0 ≤ L and k0 ≤ 0 reads �S(k0, s) = 1 s + a1 Ai(s1/3k0) + b1 Bi(s1/3k0) for 0 ≤ k0 ≤ L (43) �S(k0, s) = 1 s + a2 Ai(−s1/3k0) for k0 ≤ 0 , (44) where Ai(x) and Bi(x) are the two linearly independent solutions of the Airy differential equation U ′′(x)−xU(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Since Bi(−x) diverges as x → −∞, we discarded this in the solution for k0 ≤ 0 in equation (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The three constants (independent of k0) a1, a2, b1 are fixed by the continuity of �S(k0, s), the continuity of ∂k0 �S(k0, s) at k0 = 0 and the absorbing boundary condition �S(k0 = L, s) = 0, which yield three linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' These three constants can then be straightforwardly determined explicitly (we do not give the details here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' If the walker starts at k0 = 0 (for simplicity), from equation (44), we just need the constant a2(s) since �S(0, s) = 1 s + a2(s) Ai(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (45) It turns out that the expression of a2(s) is rather simple: a2(s) = 1 2π Ai(0) Ai′(0) s Bi(s1/3 L) = − √ 3 s Bi(s1/3 L) , (46) where we used Ai(0) = 3−2/3/Γ(2/3) and Ai′(0) = −3−1/3/Γ(1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Plugging in equation (45) then gives the exact Laplace transform, valid for all s: �S(0, s) = 1 s � 1 − 1 31/6 Γ(2/3) 1 Bi(s1/3 L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (47) Taking the Laplace transform of equation (36), and plugging in the result (47), we obtain the exact Laplace transform of the cumulative distribution of the maximum: � ∞ 0 Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≤ L] e−s t dt = 1 s � 1 − 1 31/6 Γ(2/3) 1 Bi(s1/3 L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (48) This result can be further simplified by noting that Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≥ L] = 1 − Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≤ L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Consequently, � ∞ 0 Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≥ L] e−s t dt = 1 31/6 Γ(2/3) 1 s Bi(s1/3 L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (49) Formally inverting this Laplace transform using the Bromwich contour and rescaling sL1/3 = λ, one sees immediately that for all t and L, the cumulative distribution takes the scaling form Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≥ L] = Y � t L1/3 � , (50) 11 10−1 100 m/(3t) 1 3 10−5 10−4 10−3 10−2 10−1 100 (3t) 1 3P(m) Distribution of Maximum Higher End Tail of g(z) Lower End Tail of g(z) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Distribution of the maximum of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The two full curves denote the lower end tail of g(z), equation (59), and higher end tail of g(z), equation (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The symbols are obtained from Monte Carlo simulation data for the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Starting from k0 = 0 at t = 0, the random walk was evolved up to t = 20000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The symbols show the scaled histogram obtained from n = 105000 runs of the random walk simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' where the scaling function Y (y) has the exact Laplace transform � ∞ 0 e−λy Y (y) dy = 31/3Γ(1/3) 2π 1 λBi(λ1/3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (51) While it is difficult to invert the Laplace transform exactly, it is straightforward to extract its asymptotic behaviours, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The large y behaviour of Y (y) is controlled by the small λ expansion of (51) � ∞ 0 e−λyY (y) dy ≃ 1 λ − 31/3Γ(2/3) Γ(1/3) 1 λ2/3 + 32/3Γ2(2/3) Γ2(1/3) 1 λ1/3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' , (52) which yields the large y asymptotic expansion Y (y) ∼ 1 − 31/3 Γ(1/3) 1 y1/3 + 32/3Γ2(2/3) Γ3(1/3) 1 y2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (53) The small y behaviour of Y (y) can be obtained from the large λ asymptotic behaviour of (51) � ∞ 0 e−λyY (y) dy ∼ π1/2 31/6Γ(2/3)λ11/12 e−2/3 λ1/2 , (54) which can be inverted to give the small y behaviour Y (y) ≃ 32/3 Γ(2/3)y1/3e−1/(9y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (55) 12 Using equation (50), we can now express the probability density Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) = L] of the maximum of the random walk in a scaling form: Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) = L] = − d dLProb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' [M(t) ≥ L] = 1 (3t)1/3 g � L (3t)1/3 � , (56) where the scaling function g(z) is simply related to the scaling function Y (y) and we deduce that g(z) = z−4 Y ′(y)|y=1/(3z3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (57) Using the asymptotic behavior of Y (y), we can then obtain the asymptotic tails of g(z) as g(z) ∼ 31/3 Γ(2/3) z e−z3/3 for z → ∞ (58) g(z) ∼ 32/3 Γ(1/3) � 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='32/3Γ2(2/3) Γ2(1/3) z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' � for z ≪ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (59) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (5) the tails of g(z), equations (58) and (59), are compared with the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' It is useful to compare equation (58) with equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We see that for large z, the scaling function of the position distribution (16) and that of the maximum (58) have the same asymptotic tails up to an overall factor 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' This is similar to what occurs for a simple random walk, although in that case the tails are Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The small z behaviour (59) for the scaling function is a constant with a linear correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The constant is consistent with the large time limit of the survival probability (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The linear correction contrasts with the case of a simple random walk where the correction to the constant term is quadratic in the scaling variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' GENERATING FUNCTION APPROACH In sections IV - VII, we adopted a scaling approach to obtain long-time asymptotic results for the sluggish random walk problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We now illustrate how a generating function approach may be employed to find the exact solution for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We will see that the long time limit of the solution obtained using the generating function approach recovers the results of the scaling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For the sake of brevity, we restrict ourselves to the computation of the survival probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Consider again Q(k0, t), the survival probability for a walker starting at k0 in the presence of a sink at the origin k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Q(k0, t) satisfies the backward master equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We define a generating function with parameter λ: G(k0) = ∞ � t=0 λtQ(k0, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (60) Substituting (60) into (17) and imposing the initial condition Q(k0, 0) = 1, we obtain � k0 1 − λ λ + 2 λ � G(k0) − G(k0 + 1) − G(k0 − 1) = k0 + 2 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (61) Equation (61), in which k0 takes integer values, can be solved using Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' However it is easier to take a continuum limit and expand to second order in k0, to obtain ∂2G(k0) ∂k2 0 − �(k0 + 2)(1 − λ) λ � G(k0) = −(k0 + 2) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (62) The homogeneous version of (62) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' equating the lhs to zero) has Airy functions as solutions: Ghom(k0) = B0Ai(C(k0 + 2)) + B1Bi(C(k0 + 2)) , (63) where C = �1 − λ λ �1/3 , (64) 13 and the constants B0 and B1 are to be fixed by the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We can set B1 = 0 and discard the Bi solution, as it diverges as k0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Then for G(k0) a particular solution to (62) is 1/(1 − λ) and the general solution to (62) is G(k0) = B0Ai((k0 + 2)C) + 1 1 − λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (65) The boundary condition is G(0) = 0, which fixes the constant B0, and we obtain the solution to (62) as G(k0) = 1 1 − λ � 1 − Ai((k0 + 2)C) Ai(2C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (66) We are interested in the long time asymptotic behaviour, which we can extract from the λ → 1 limit of (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Since C → 0 as λ → 1, we require the small argument expansion of the Airy function: Ai(x) ≃ 1 32/3Γ(2/3) − x 31/3Γ(1/3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (67) We then find, in the limit λ → 1, G(k0) ≃ 31/3 Γ(2/3) Γ(1/3) k0 (1 − λ)2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (68) Thus the leading singularity is at λ∗ = 1 and is of the form (λ∗ − λ)−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Invoking the usual Tauberian theorem [33], this singularity gives the following large t asymptotic behaviour: Qt(k0) ∼ 31/3 Γ(1/3)k0t−1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (69) This matches perfectly with the small z asymptotic of the scaling behaviour in (24) upon using the small z expansion of f(z) in equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' GENERALISATION TO THE CASE WHERE α ̸= 1 Up to now, we have considered only the case where the probability of hopping to the right or left is proportional to 1/(|k| + 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' the exponent α = 1 in the general expression for the hopping probability, A(|k| + 2)−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We now generalise to the case where α ̸= 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' the hopping probability is proportional to 1/(|k| + 2)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In this case equation (3) generalises for k ≥ 0 to ∂ ∂tP(k, t) ≈ 1 kα � ∂2 ∂k2 P(k, t) − 2α k ∂ ∂k P(k, t) + α(α + 1) k2 P(k, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (70) Equation (70) can be put into the standard form (6), where now D(k) = 1/kα and U(k) = 1/kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' One can again solve (70) by the scaling approach discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For general positive α it is easy to show that the scaling variable becomes k/tν where ν = (2 + α)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Therefore the solution of (70) for P(k, t) has a scaling form P(k, t) = t− 1 α+2 G � k t− 1 α+2 � , (71) where the scaling function G(z) is symmetric and, for positive z, satisfies the nontrivial differential equation G′′(z) + � zα+1 α + 2 − 2α z � G′(z) + � zα α + 2 + α(α + 1) z2 � G(z) = 0 , (72) with boundary condition G(z) → 0 as z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Remarkably, this equation admits the simple solution, satisfying the boundary condition, G(z) = A zα exp � − zα+2 (α + 2)2 � , (73) 14 where the normalisation constant A is given by A−1 = 2(α + 2) α α+2 Γ �α + 1 α + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (74) Using the symmetry G(z) = G(−z), the full solution for all z can be written as G(z) = A |z|α exp � − |z|α+2 (α + 2)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (75) When α = 0 we recover the standard Gaussian result for a simple random walk, while for α = 1 we recover the result (16) upon rescaling z → 31/3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We note that for any α > 0 there is a trough, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' a cusp singularity, at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The trough at z = 0 disappears only for the case of simple diffusion (α = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Similar scaling analyses can be performed for the survival probability as well as the distribution of the maximum site visited to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We do not repeat the analysis, but just note that the scaling implies that the asymptotic decay of the survival probability is Q(t) ∼ t−1/(α+2) and the maximum scales as M(t) ∼ t1/(α+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' CONCLUSION In this paper we have studied a random walk with space-dependent transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Our study was motivated by trap models of slow dynamics, but in contrast to most such models, our trap depths are not random but instead increase logarithmically with distance k from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The dynamics of a particle moving on the lattice of traps follows an inhomogeneous random walk which has symmetric transition probabilities that decrease with k as 1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Thus the motion of a walker slows down as it goes further and further away from the origin, a phenomenon that we term ‘sluggish dynamics’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The sluggish dynamics causes the typical distance explored up to time t to grow subdiffusively as t1/3, in contrast to the standard t1/2 law for a simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We used a scaling approach, in which the scaling variable is k/t1/3, to compute long-time asymptotic results for various properties of this inhomogeneous random walk: the position distribution, the survival probability in the presence of a sink at the origin, the joint survival and position distribution, and the distribution of the maximum distance to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Interestingly, the position distribution has a trough (a cusp singularity) at the origin and is bimodal, with two peaks located at |k| = (3t)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The contrasts with the usual Gaussian distribution for diffusion (which has a single maximum at k = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The bimodal distribution and the t1/3 scaling reflect the sluggish nature of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The survival probability shows an asymptotic decay ∼ t−1/3 at large time, which contrasts with the t−1/2 decay for a simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The fact that the survival probability decays to zero as t → ∞ implies that the walk is recurrent in d = 1, as is the simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The distribution of the maximum of the walk up to time t has a nontrivial scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' We further showed how a generating function approach can be used to find exact solutions for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Using this approach to compute the survival probability in the presence of a sink at the origin, we recover our scaling result in the long-time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Application of the same generating function approach to other observables should be a straightforward extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Finally, we generalised the model to cases where the transition probability decays as 1/|k|α with positive α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Except for α = 0 (simple random walk), the position distribution always shows a trough at the origin (k = 0), where it exhibits a singularity, behaving as |k|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' Remarkably, the scaling function for the position distribution takes on a simple form (equation (75)) and there is always a trough at the origin with associated singularity |z|α for α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' It is worthwhile comparing the behaviour of our sluggish random walk model with that of the Gillis model outlined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In the continuum limit the Gillis model becomes diffusion in a logarithmic potential [19, 24] and the corresponding Fokker-Planck equation reads ∂ ∂tP(k, t) = ∂ ∂k � ∂ ∂k P(k, t) + � ∂ ∂k U(k) � P(k, t) � , (76) where the potential U(k) = 2ϵ ln |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The relevant case for us is ϵ < 0 whereby the potential is repulsive and the particle is pushed away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' In this Gillis case, the solution for the time-dependent position distribution has scaling form [19, 24] P(k, t) → 1 t1/2 GGill � k t1/2 � (77) 15 where the scaling function, GGill(z), is given by GGill(z) = 2ϵ−1/2 Γ(1/2 − ϵ) |z|−2ϵ e−|z|2/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' (78) This is to be compared with the scaling function G(z) (16) for the sluggish random walk model (where the scaling variable is z = k/(3t)1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' As with (16), the scaling function (78) is bimodal, with peaks at z = ±(−2ϵ)1/2, and has a trough at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' However, the model exhibits diffusive scaling and is thus not sluggish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The difference between the sluggish random walk and diffusion in a logarithmic potential is evident when one compares the Fokker Planck equations (6) and (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The key difference is the space-dependent diffusion constant D(k) = 1/k appearing in (76), along with the potential U(k) = 1/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' It is these features that lead to a change of the scaling variable to z = k/(3t)1/3 and consequent sluggish behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' The sluggish random walk model and its analysis are straightforward to generalise to higher dimensions and other observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For example, it would interesting to study the return probabilities and recurrence/transience transition in a higher dimension for general α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' It would also be of interest to study the time for the walker to traverse from one maximum of the position distribution to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' More generally our study has shown that inhomogenous space-dependent random walks can exhibit surprising properties and it remains to explore the full range of such behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' AZ acknowledges support of the INSPIRE fellowship from DST India and the Physics Computing Facility lab at UCSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' RJA was supported by the European Research Council under consolidator grant 682237 EVOSTRUC and by the Excellence Cluster Balance of the Microverse (EXC 2051 - Project-ID 390713860) funded by the Deutsche Forschungsgemeinschaft (DFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFPT4oBgHgl3EQfZTSq/content/2301.13077v1.pdf'} +page_content=' For the purpose of open access, the author has applied a Creative Commons Attribu- tion (CC BY) licence to any Author Accepted 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--git a/.gitattributes b/.gitattributes index d436a9c9f51a6ccec39a2e83e37c2e080f590da1..07c9b8d739fe1224abf07beac05ed612c01f9bef 100644 --- a/.gitattributes +++ b/.gitattributes @@ -184,3 +184,4 @@ tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf filter=lfs diff=lfs merge=lfs -tex V9AzT4oBgHgl3EQfJ_vP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text x9FJT4oBgHgl3EQfhSy6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf filter=lfs diff=lfs merge=lfs -text +o9FMT4oBgHgl3EQf7jFB/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/29AzT4oBgHgl3EQffPxu/content/tmp_files/2301.01449v1.pdf.txt b/29AzT4oBgHgl3EQffPxu/content/tmp_files/2301.01449v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eeb5d402595427c4e4fbdf4ebcf47b418deccabf --- /dev/null +++ b/29AzT4oBgHgl3EQffPxu/content/tmp_files/2301.01449v1.pdf.txt @@ -0,0 +1,916 @@ +Building Coverage Estimation with Low-resolution Remote Sensing Imagery +Enci Liu, Chenlin Meng, Matthew Kolodner, Eun Jee Sung, Sihang Chen +Marshall Burke, David Lobell, Stefano Ermon +Stanford University +{chenlin, jesslec}@cs.stanford.edu, {mkolod, ejsung, schen22, mburke, dlobell}@stanford.edu, ermon@cs.stanford.edu +Abstract +Building coverage statistics provide crucial insights into the +urbanization, infrastructure, and poverty level of a region, fa- +cilitating efforts towards alleviating poverty, building sustain- +able cities, and allocating infrastructure investments and pub- +lic service provision. Global mapping of buildings has been +made more efficient with the incorporation of deep learning +models into the pipeline. However, these models typically +rely on high-resolution satellite imagery which are expensive +to collect and infrequently updated. As a result, building cov- +erage data are not updated timely especially in developing re- +gions where the built environment is changing quickly. In this +paper, we propose a method for estimating building coverage +using only publicly available low-resolution satellite imagery +that is more frequently updated. We show that having a multi- +node quantile regression layer greatly improves the model’s +spatial and temporal generalization. Our model achieves a co- +efficient of determination (R2) as high as 0.968 on predicting +building coverage in regions of different levels of develop- +ment around the world. We demonstrate that the proposed +model accurately predicts the building coverage from raw in- +put images and generalizes well to unseen countries and con- +tinents, suggesting the possibility of estimating global build- +ing coverage using only low-resolution remote sensing data. +Introduction +The quantity and location of buildings provide important in- +sight into the human activities and urban development of +a region. Not only are building statistics themselves key +socioeconomic indicators, they also help predict other key +sustainable development indices, including poverty (Ayush +et al. 2021; Uzkent, Yeh, and Ermon 2020; Yeh et al. 2020), +population density (Huang et al. 2021), and climate out- +comes (Chini et al. 2018). Moreover, building coverage +statistics help policymakers and NGOs make informed deci- +sions regarding the provision of public services, the targeting +of humanitarian aid, and priorities for large-scale infrastruc- +ture investments. +The development of deep learning detection (Redmon and +Farhadi 2018) and segmentation models (Ronneberger, Fis- +cher, and Brox 2015; Sun et al. 2019) has allowed for more +efficient global mapping of buildings. As a result, there has +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +been an increasing number of global settlement map datasets +in the past decade (Esch et al. 2017; Marconcini et al. 2020; +Sirko et al. 2021), allowing researchers to develop insights +into the socioeconomic development of different regions. +Nevertheless, detection- or segmentation-based methods +typically rely on a large amount of high-resolution satel- +lite imagery and the corresponding pixel- or instance-level +labels for training, which are often prohibitively expensive +and unaffordable for researchers and policymakers. More- +over, high-resolution imagery are updated less frequently +than low-resolution ones. In addition, running detection or +segmentation models over high-resolution images covering +a large area requires a large amount of compute. Due to +these reasons, building data gathered in this way is often not +updated timely. For example, the Microsoft Global Build- +ing Footprints dataset1 is collected from satellite images be- +tween 2014 and 2021. However, during the 7-year period, +population continues to grow and new buildings are con- +structed, especially in fast developing regions. For instance, +from 2015 to 2020, the population increased by 44.1% +for Malappuram and 34.2% for Abuja.2 Moreover, fine- +grained detection and segmentation models usually gener- +alize poorly to unseen geographies, timestamps, and image +sources because the appearance of buildings vary widely in +satellite images (Yuan and Cheriyadat 2014). +Compared with its high-resolution counterpart, low- +resolution satellite imagery (e.g. Sentinel-1 and Sentinel- +2) are publicly available and updated every month, making +them desirable for studying the economic and urban devel- +opment of a region. However, prior works have not fully +utilized low-resolution imagery. In this paper, we propose +a cheaper and more generalizable way to update building +statistics using only low-resolution satellite imagery from +Sentinel-1 and Sentinel-2. Instead of detecting or segment- +ing each building in satellite images, the proposed model +directly predicts the building coverage from the raw input +pixels, using input imagery from a public source that is up- +dated nearly weekly. Specifically, we found that incorporat- +ing a multi-node quantile regression loss helps improve the +generalization of the model. The proposed method achieves +a coefficient of determination (R2) as high as 0.968 in re- +1https://github.com/microsoft/GlobalMLBuildingFootprints +2https://worldpopulationreview.com/ +arXiv:2301.01449v1 [cs.CV] 4 Jan 2023 + +gions from different continents and of different levels of de- +velopment. We also conduct ablation studies and show that +the incorporation of additional multi-spectral bands avail- +able from the low-resolution satellite imagery and the multi- +node quantile regression design help improve the model per- +formance. +Method +In this paper, we propose a deep-learning based regression +model that accurately predicts building coverage in low- +resolution satellite imagery from Sentinel-1 and Sentinel- +2. Unlike detection- or segmentation-based methods, our +model does not require high-resolution training data or +hand-crafted threshold and generalizes well to unseen re- +gions. The proposed method accurately predicts the building +coverage in the raw input low-resolution satellite image with +the help of a quantile regression. +Problem Definition +Given a geographical region, we want to estimate the build- +ing coverage within it. Due to the lack of direct statistics of +building coverage in square meter or kilometer, we instead +predict the number of building pixels y ∈ R, in the satellite +image X representing the target region, assuming that the +number of building pixels is proportional to the actual build- +ing coverage (Figure 1(a)). We want to build a model that +predicts y from the raw image input X. +Multi-node Quantile Regression Model +We observe that the distribution of building pixel counts +across Africa and South America are heavy-tailed, with +more than 75% of the samples having less than 20% of all +pixels being buildings. Regular linear regression trained on +root-mean-square error objective estimates the conditional +mean and assumes normality of the data distribution, fail- +ing to model non-normal asymmetric distribution accurately. +Moreover, linear regression is not robust to outlier values, +which characterize the building pixel count distribution for +our task. +To address the aforementioned problems, we adopt quan- +tile regression (Koenker and Bassett Jr 1978), which esti- +mates the conditional quantile (e.g., median) of the response +variable. Quantile regression allows us to incorporate uncer- +tainty in prediction and captures the relationship between the +input and different quantiles of the data. As we will see in +Section , multi-node quantile regression indeed empirically +performed better than regular linear regression for our task. +Specifically, we modify the ResNet18 (He et al. 2016) ar- +chitecture to have K output channels (i.e. nodes), each cor- +responding to a different quantile (see Figure 1). As it is ob- +served that the median of a distribution gives the minimum +absolute error from the ground truth (Hanley et al. 2001), at +inference time, we collect the model predictions from only +the 0.5 quantile node, which is expected to predict the con- +ditional median of the response variable. +Multi-node Quantile Loss +For each node that represents a quantile q ∈ (0, 1), we com- +pute an asymmetric quantile loss, or pinball loss. Depending +on the quantile q, over- and under-estimation are penalized +unevenly. Specifically, the node-wise pinball loss for a given +prediction ˆy and ground truth label y is computed as follows: +Lpinball(q, y, ˆy) = +�q · |ˆy − y| +if ˆy ≥ y +(1 − q) · |ˆy − y| +otherwise +When q = 0.5, the pinball loss is the same as the absolute +error. +To compute the final loss, we take the mean of the pinball +losses among the K output nodes as follows: +Lquantile(y, ˆy) = 1 +K +K +� +n=1 +Lpinball(qn, y, ˆy) +where the subscript n indicates the index of the quantile in +the list of quantiles predicted by the model. Notice that when +K = 1 and q = 0.5, the quantile loss formula boils down to +the regular L1 loss. +Experiment Setup +Before introducing the experiment setups, we define tile and +patch in the context of this paper. We define a patch to be the +small rasters of 50×50 pixels that we crop larger rasters into. +A tile is a larger satellite imagery that could be cropped into +multiple smaller patches (see Figure 2). We train and eval- +uate the multi-node quantile regression model on patches of +50 × 50 pixels and add post-processing steps to collect tile- +level results. +Training Data +As the majority of the African continent is covered by for- +est or desert and does not contain any buildings, we sample +locations based on the population density so that the train- +ing data contains sufficient tiles with buildings for the model +to learn from. Our training set contains 15,000 input-label +pairs. Sentinel-1 and Sentinel-2 images are used as as the +input data and the Open Buildings dataset is used to derive +the building coverage label, as we detail next. +Sentinel-1 (S1) +satellites collect radar imagery for land +and ocean monitoring. We download the S1 satellite im- +agery collected in 2020 from Google Earth Engine. The im- +ages have 10m GSD and are composites that take the median +value over the target period of time. We include band 1, the +VV overall mean, as one of the input channels. The S1 tiles +are cropped into 50 × 50-pixel patches. As areas with high +built-up density are shown to result in stronger signals in +S1 band 1 (Koppel et al. 2017), we expect that adding this +channel to the input will improve the model’s performance. +Sentinel-2 (S2) +satellites are equipped with the mission of +land monitoring. We download the 2020 composites of S2 +imagery from Google Earth Engine. We include the RGB +channels (i.e. bands 4, 3, 2) and the near-infrared (NIR) +channel (i.e. band 8) from S2 as input. Compared with other +channels, NIR is useful for distinguishing the vegetation +from the buildings (Luo et al. 2019; Pessoa et al. 2019; +Schlosser et al. 2020). All of the four bands are available +in 10m GSD. The S2 tiles are cropped into 50 × 50-pixel +patches. + +(a) Predict the number of building pixels 𝑦 in image 𝑋 +𝑦 = 874 +Base map (left) and the binary mask +(right) used to derive the label 𝑦 +0.1 +0.9 +Concat +ResNet18 +(b) Multi-node quantile regression model +#𝑦 +0.5 +ℒ!"#$%&'( +𝐾 = 3 +Input 𝑿 +𝐴 = 34.96% +Figure 1: (a) Given an input image X, we want to predict the number of building pixels y within it. We assume that y ap- +proximate the actual building coverage. (b) Architecture of the multi-node quantile regression model. The input to the model +includes five channels, which are Sentinel-1 band 1 (Overall Mean), Sentinel-2 band 4, 3, 2, 8 (R, G, B, NIR). The model has +K output nodes representing different quantiles. +Tile +Patch +Crop +Summation +50 +50 +Figure 2: A tile can be cropped into multiple 50 × 50 pix- +els patches. The tile-level results are computed by taking +summation of all the patch-level predictions. +Open Buildings +(Sirko et al. 2021) contains 516M build- +ing footprints across 43 African countries that cover 64% +of the continent. The building footprints were detected us- +ing state-of-the-art segmentation model collected from high- +resolution imagery at different timestamps. We downsample +the original high-resolution mask (0.5m GSD) to 10m GSD +to match the resolution of the input data. Then, we convert +the continous-valued rasters into binary masks by threshold- +ing at 0. The building pixel count labels are derived for the +50 × 50-pixel patches. +Experiment Settings +To evaluate the generalization performance of the model, we +consider three experiment settings that captures likely cases +of real-world application. +Holistic. +In the holistic setting, we train and test on data +points sampled across the African continent based on popu- +lation density. +Intra-country. +In the intra-country setting, we train and +test the model on samples from the same country. +Exclusive. +In the exclusive setting, we train the model on +all African countries except for the one country on which we +test our model. +Baselines +To evaluate the performance of the proposed method, we +use existing settlement map products from different years +as baselines to compare against. These off-the-shelf prod- +ucts provide researchers and policymakers with general in- +formation about urban shapes and boundaries, but could be +less useful in providing up-to-date building coverage statis- +tics, which change more frequently than the shape of urban +area. We believe that these existing settlement map products +serve as good baselines to compare our method against and +provide information of them in this section. +Gloal Urban Footprint (GUF) +GUF was collected from +2011 to 2012. It contains mappings of human settlements in +the form of binary masks. +Global Human Settlement Layer (GHSL) +GHSL was +collected from Sentinel-1 images in 2018. The building map +is available as binary masks. +World Settlement Footprint (WSF) +WSF (Marconcini +et al. 2020) was collected in 2015. It contains binary masks +of global human settlements. +Evaluation Settings +We evaluate the model performance at both patch-level and +tile-level and describe the evaluation metrics in this section. +Patch-level Evaluation +As the model is trained on +patches, we want to evaluate the model’s performance at the +same scale. We use two evaluation metrics for patch-level +evaluation: mean absolute error (MAE) and Pearson’s r2 be- +tween the predicted and the ground truth labels. +Tile-level Evaluation +In real-world applications, building +coverage statistics are needed over a large geography. To re- +flect this use case, we also evaluate the model performance at + +1661 (1687) +82 (83) +93 (87) +665 (645) +698 (704) +987 (945) +Figure 3: Examples of model predictions in Brazil, which was unseen during training. The first row is the RGB input image; the +second row is the binary masks (where the bright yellow pixels are building pixels) from which we derive the label. The model +prediction is in black; the ground truth labels are highlighted in green in the parentheses. Our method accurately estimated the +results. +Expt./Eval. settings* +Open Buildings +SpaceNet7 +Holistic +✓ +Intra-country +✓ +Exclusive +✓ +Patch-level +✓ +✓ +Tile-level +✓ +Table 1: Checkmarks indicate that the corresponding dataset +is used as validation data under the experiment (Expt.) or +evaluation (Eval.) settings. *Different experiment settings +require retraining the model, while different evaluation set- +tings do not. +tile-level using absolute error in building coverage. To com- +pare the statistics of building coverage across baselines with +different GSDs, we compute the percentage of building pix- +els within a tile as a proxy for building coverage. For a tile +R cropped into N patches at inference time, let yi be the +number of building pixels in patch i, the building coverage +percentage is computed as: +Cbuilding(R) = +�N +i=1 yi +Htile × Wtile +× 100 +where Htile and Wtile denote the height and width (in pixel) +of the tile. The absolute error between a method and the +ground truth is computed as the absolute difference between +the two building coverage percentages. +Evaluation Data +We evaluate our model on Open Buildings and SpaceNet 7 +Challenge datasets and provide the information in Table 1. +Open Buildings +The Open Buildings (Sirko et al. 2021) +dataset provides the training labels. We evaluate the model +performance on a hold-out test subset of the Open Buildings +dataset at patch-level only. We do not use Open Buildings for +tile-level evaluation because it contains data from different +timestamps. +SpaceNet 7 Challenge dataset (SpaceNet7) +SpaceNet73 +was published in 2020 as the data for the SpaceNet 7 Multi- +Temporal Urban Development Challenge. The dataset pro- +vides 4km × 4km tiles and the building polygons in each +tile. We downsample the raster to 10m GSD to match the +resolution of the input data. We evaluate the model perfor- +mance on SpaceNet7 at both patch-level and tile-level. +We use the SpaceNet7 tiles for validation because the la- +bels were collected the same year as the Sentinel-1/-2 in- +put data. Furthermore, SpaceNet7 includes tiles from re- +gions outside of Africa, allowing us to evaluate the model’s +performance on unseen countries. Specifically, we evaluate +our model on six SpaceNet7 tiles from different regions: +Uganda, Zambia, Ghana, Peru, Brazil, and Mexico. Note +that we do not use SpaceNet7 as the training labels be- +cause the data is limited in quantity. These labels are human- +generated and likely of higher quality compared to those +from Open Buildings, which are generated by a model. +Results +In this section, we evaluate the performance of the proposed +method. We first show that our model accurately predicts +building coverage in Africa and generalizes to South Ameri- +can regions unseen during training. We also conduct ablation +studies demonstrating that the major design choices – non- +RGB bands and multi-node quantile regression – are neces- +sary for boosting model performance and generalization. +3SpaceNet on Amazon Web Services (AWS). “Datasets.” The +SpaceNet Catalog. Last modified October 1st, 2018. Accessed on +November 20th, 2021. https://spacenet.ai/datasets/ + +Africa +South America +Uganda +Zambia +Ghana +Brazil +Peru +Mexico +Method +R2 ↑ +Tile ↓ +R2 ↑ +Tile ↓ +R2 ↑ +Tile ↓ +R2 ↑ +Tile ↓ +R2 ↑ +Tile ↓ +R2 ↑ +Tile ↓ +GUF (2012) +0.092 +7.23 +-0.715 +0.96 +0.466 +1.59 +– +– +– +– +– +– +WSF (2015) +0.286 +0.21 +-5.444 +38.68 +-0.290 +3.28 +0.579 +9.21 +0.562 +6.06 +-0.099 +18.52 +GHSL (2018) +0.057 +6.83 +0.023 +2.46 +0.771 +0.75 +0.863 +0.97 +0.516 +10.06 +0.187 +8.12 +w/o multi-node QR +-15.51 +33.15 +-10.41 +54.27 +-15.64 +31.68 +-0.069 +18.30 +-2.807 +38.28 +-2.480 +36.24 +w/o S1 band 1 +0.809 +1.95 +0.779 +3.74 +0.252 +3.37 +0.864 +3.91 +0.906 +2.34 +0.422 +10.96 +w/o S2 band 8 +0.772 +2.33 +0.580 +7.14 +0.804 +0.54 +0.751 +6.19 +0.836 +3.13 +0.337 +13.36 +Ours +0.868 +1.18 +0.866 +3.21 +0.835 +2.31 +0.968 +0.83 +0.798 +6.63 +0.707 +0.36 +Table 2: Results on SpaceNet7. The bottom four rows are the proposed methods with the corresponding components removed +and the complete version. In the table, r2 is the patch-level Pearson’s r2 and Tile is the tile-level absolute error between +SpaceNet7 and the corresponding method. The model was trained with 15,000 samples in Africa for 1200 epochs and tested on +the corresponding SpaceNet7 tiles. +Expt. setting +Train +Test +MAE ↓ +R2 ↑ +Holistic +Africa +Africa +75.43 +0.888 +Intra-country +Rwanda +Rwanda +27.60 +0.938 +Exclusive +Africa* +Rwanda +42.04 +0.844 +Exclusive +Africa* +Uganda +207.74 +0.568 +Excluisve +Africa* +Kenya +110.67 +0.915 +Table 3: Patch-level results for different experiment settings +on Open Buildings. All models in the table are trained with +15,000 samples from the corresponding train regions for +1200 epochs and tested on 1,000 samples from the corre- +sponding test regions. *Under the Exclusive setting, the cor- +responding test region is removed from the training set so +that the model is tested on unseen regions. +Accurate Prediction in Africa +To demonstrate the effectiveness of our method, we evaluate +it on both SpaceNet7 (see Table 2) and Open Buildings (see +Table 3) in Africa. We define a tile as multiple patches and +compute the tile-level results by taking summation of the +patches within it (see Section 4.1). +As shown in Table 2, our model achieves an R2 as high as +0.968 at patch-level, outperforming the baselines on most of +the African regions. Some examples of the proposed model’s +patch-level prediction are provided in Figure 3. Furthermore, +the proposed method yields fairly accurate building cov- +erage estimates at tile-level, achieving a low error rate of +0.54% on Ghana. Additionally, we observe that baselines +like WSF and GUF give good estimates at only tile-level +but not the other, indicating that the errors at patch-level are +cancelled out. In contrast, the proposed method yields con- +sistently accurate estimations at both tile- and patch-level. +We emphasize that having good performance on both scales +is ideal, since it gets us closer to finer-grained pixel-level +predictions (semantic segmentation). +Generalization to Unseen Regions +Prior methods for generating building coverage statistics +generalize poorly under domain shift. To evaluate the gen- +eralizability of the proposed method, we train and test the +multi-node quantile regression model under three exper- +iment settings (defined in Section 4.3) – Holistic, Intra- +country, and Exclusive. We provide the patch-level results +on the Open Buildings dataset in Table 3. +We see that among the three experiment settings, Intra- +country gives the highest Pearson’s r2 of 0.971 because the +model is tested on in-domain data and the region is small. +We also observe that the proposed multi-node quantile re- +gression model generalizes well to regions not seen during +training. Speicifcally, under the Exclusive setting, the pro- +posed method achives an r2 as high as 0.962 even with the +test regions removed from the training set (see Table 2). +Furthermore, the proposed model generalizes to regions +outside of the African continent. We evaluate our method +on SpaceNet7 tiles from South America and provide the +results in Table 2. We observe that the proposed model +achieves comparable or superior performance when evalu- +ated on Brazil, Peru, and Mexico compared with the base- +lines. This generalization to a different continent indicates +that our model could potentially be applied to the globe for +collecting building coverage statistics, while using only pub- +licly available low-resolution satellite imagery. +Ablation Studies +In this part, we carry out ablation studies on 1) the incorpo- +ration of S1 band 1 and S2 band 8 as input and 2) the multi- +node quantile regression, and provide the results in Table 2. +Multi-node Quantile Regression +To see the effect of hav- +ing multiple output nodes for different quantiles, we com- +pare the performance of the multi-node quantile regression +model with the single-node model trained on the L1 objec- +tive. We provide the ablation study results on SpaceNet7 in +Table 2 (see row “w/o multi-node QR”). We observe that +the proposed multi-node model outperforms the single-node +model by a large margin on all test regions. In addition, as +shown in the scatter plots for ground truth VS. predicted +building pixel counts (Figure 4), the model tends to overesti- +mate the building coverage without the multi-node quantile +regression. We speculate that having multiple output nodes +improve performance because the pinball losses from the 0.1 +and 0.9 quantile nodes help bound the predictions. +Multi-Spectral Bands +We conduct ablation studies on the +two non-RGB multi-spectral bands — i.e. S1 band 1 and S2 + +w/o S2 band 8 +w/o S1 band 1 +w/o Multi-node QR +Ours +𝑹𝟐 = 0.751 +𝑹𝟐 = 0.864 +𝑹𝟐 = -0.069 +𝑹𝟐 = 0.968 +Ground truth building pixel count +Predicted building pixel count +Figure 4: Scatter plots of patch-level predicted number of +building pixels against the ground truth from the ablation +studies. All models are trained on Africa and tested in the +Brazil tile. Removing the multi-node quantile regression or +any of the non-RGB bands makes the model be prone to +overestimation or underestimation. +band 8 — and provide the ablation study results evaluated on +SpaceNet7 in Table 2 . We observe that at both patch- and +tile-level, incorporating S1 band 1 and S2 band 8 boosts the +performance. From the scatter plots in Figure 4, we see that +removing either S1 band 1 or S2 band 8 makes the model un- +derestimates at patch-level. This suggests that the non-RGB +bands provide additional information that helps correct the +model’s tendency to under- or over-estimate. +Discussion and Social Impact +United Nations’ Sustainable Development Goals (SDGs) +present an urgent call for action in all countries and collabo- +ration between different sectors of policy-making for a more +sustainable development (Nations 2016). However, relevant +data for informing the decision makers and organizations are +often lacking or infrequently collected, especially in fast de- +veloping countries. Building coverage is an important so- +cioeconomic indicator and also helps predict other indica- +tors. In this paper, we develop a framework for estimat- +ing building coverage using only free low-resolution satel- +lite imagery from Sentinel-1 and Sentinel-2. The proposed +multi-node quantile regression model yields fairly accurate +estimates and generalizes well to unseen countries and con- +tinents. +This paper offers a cost-efficient and generalizable way +to collect global building coverage statistics, accelerating +the progress towards multiple SDGs. For example, build- +ing coverage helps predict the level of economic develop- +ment, which informs policymakers’ decisions for alleviating +poverty (SDG 1 No Poverty), allocating infrastructure re- +sources (SDG 9 Industry, Innovation and Infrastructure), and +building more sustainable cities (SDG 11 Sustainable Cities +and Communities). In addition, building coverage provides +important information about the interaction between human +and environment, including the monitoring of agriculture to +reduce hunger (SDG 2 Zero Hunger) and climate measure- +ment (SDG 13 Climate Action). +Furthermore, the proposed method could potentially be +applied to track changes of building coverage for different +regions in the world, assisting existing efforts for this. For +example, United Nations’ World Urbanization Prospects re- +port provides estimates and projections of urban and rural +data, including population and area, throughout the time +(Nations 2018). The proposed method could be applied to +derive building coverage statistics as soon as satellite images +are updated (the low-resolution satellite imagery are updated +on a weekly basis). As building coverage is highly corre- +lated with or can be used to derive other values like building +density, urban area, and population, our method could po- +tentially aid the efforts to track urban development. +This paper demonstrates the viability of using low- +resolution free satellite imagery for estimating building cov- +erage statistics over a large geography. Future research could +explore how the model can be applied to classify urban and +rural areas, and estimate population and poverty levels. +References +Ayush, K.; Uzkent, B.; Tanmay, K.; Burke, M.; Lobell, D.; +and Ermon, S. 2021. Efficient Poverty Mapping from High +Resolution Remote Sensing Images. In AAAI. +Chini, M.; Pelich, R.; Hostache, R.; Matgen, P.; and L´opez- +Mart´ınez, C. 2018. 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Learning to count +buildings in diverse aerial scenes. +In Proceedings of the +22nd ACM SIGSPATIAL International Conference on Ad- +vances in Geographic Information Systems, 271–280. + +Appendix A: Datasets +Sentinel-1 +provides free global Synthetic Aperture Radar +(SAR) imagery available with 9 bands at ground sampling +distance (GSD) of 10m. The revisit time of the Sentinel-1 +satellite is 12 days, meaning that the images are updated fre- +quently. We include band 1, the VV overall mean, as one of +the input channels. Each Sentinel-1 tile that we download is +2km-by-2km and has the shape 200×200 pixels. Each tile is +then cropped into 16 smaller patches of shape 50×50 pixels +when fed into our model. +Sentinel-2 +provides multi-spectral images in from the vis- +ible to the shortwave infrared spectral range (SWIR) with the +revisit time of 10 days. The Sentinel-2 satellite imagery con- +tains 13 multi-spectral bands with GSDs ranging from 10 to +60m. Besides the RGB channels, we also included the near- +infrared (NIR) channel (i.e. band 8) as input. Like Sentinel- +1, all the Sentinel-2 imagery are downloaded as 2km-by- +2km tiles and cropped into 16 patches of size 50 × 50-pixel. +Open Buildings +contains 516M building footprints across +43 African countries that cover 64% of the continent. The +building footprints were detected using state-of-the-art seg- +mentation models. The Open Buildings data is originally in +the format of polygons labeled with three confidence inter- +vals of buildings presence - 0.6 to 0.65, 0.65 to 0.7, and +greater than 0.7. For our purpose, we download the Open +Buildings data as high-resolution rasters with 0.5m GSD and +two bands. Band 1 demonstrates the model confidence that a +building is located in the region with the confidence interval +preserved from the original polygon data. A band 1 value of +zero indicates no building presence. Band 2 is a reclassifica- +tion of the confidence scores into four buckets. +We use band 1 to derive the binary label of building and +non-building. Specifically, we first downsample the rasters +to 10m GSD to match the resolution of the Sentinel-1 and +Sentinel-2 input images. Then, we convert the continous- +valued band 1 into a binary mask by treating all pixels with a +non-zero value as a building pixel – i.e. a pixel that contains +buildings. Like Sentinel-1 and Sentinel-2, the downsampled +binary mask is cropped into 50 × 50-pixel patches. For each +smaller patch, we use the number of building pixels as the +labels for training and testing. +Appendix B: Baselines +Figure 5 shows the visualization of different benchmarks de- +scribed below and datasets we used in experiments. +Gloal Urban Footprint (GUF) +is a worldwide mapping +of human settlement patterns in the form of binary masks +available in 12m GSD. The building footprints are derived +from satellite images of TerraSAR-X and TanDEM-X from +2011 to 2012. +Global Human Settlement Layer (GHSL) +was collected +from backscattered information of Sentinel-1 images. The +building map is available as binary masks at 20m GSD. +World Settlement Footprint (WSF) +is a binary mask of +global human settlements available in 10m GSD. The map +Base map +Open Buildings (2021) +WSF (2015) +GHSL (2018) +GUF (2012) +SpaceNet (2020) +Figure 5: Binary settlement maps in Zambia (1 = red build- +ing pixel; 0 = white non-building pixel) from different +sources. The years of collection for the settlement maps are +provided in the parentheses. +(a) Geo-locations of training samples +(b) Per km2 cost and resolution of +different satellite imagery +Figure 6: (a) The geo-locations of our training samples, all +of which are from Africa. (b) High-resolution satellite im- +agery are expensive, while many lower-resolution ones are +publicly available. +was derived from Sentinel-1 and Landsat-8 satellite imagery +in year 2015. +Appendix C: Experiments +Model Implementation +We modify the ResNet18 architecture to incorporate multi- +node quantile regression. Specifically, we replace the final +fully-connected (FC) layer in ResNet18 with two FC layers, +each followed by a ReLU activation. We set the number of +output channels (i.e. nodes) to be K, which is the number +of quantiles the model predicts. For all models in this pa- +per, we use K = 3 and predict the quantiles {0.1, 0.5, 0.9}. +Each channel corresponds to a quantile and the correspond- +ing pinball loss is computed using the output of that partic- +ular channel. To prevent the model from overfitting, we add +dropout layers after each ResNet block, which we found em- +pirically that it helps the model generalize to unseen regions. + +Price($/km^2) +GSDmRegion +Population (%) +Building (%) +Dhaka +19.23 +29.68 +Kampala +28.18 +7.28 +Athens +-0.19 +1.55 +Boston +4.45 +-9.42 +Table 4: Temporal change experiment results on four cities +of different levels of development. The second and third are +percentage change from 2016 to 2021. All results are col- +lected from the model trained with 15,000 samples in Africa +for 1200 epochs. +Training Details +The training samples’ geo-locations are shown in Figure 6 +(a). For all models in the experiment section, we train them +on 15000 samples for 1200 epochs with a learning rate of +0.002. The models have fully converged at the point where +we end training. +Temporal Experiments +Evaluation +To evaluate our model’s ability to track tem- +poral changes in building coverage, we run the model on +satellite images from 2016 and 2021.4 Specifically, we chose +four cities with various levels of development from four dif- +ferent continents – Dhaka, Kampala, Athens, and Boston – +and downloaded all images covering the cities. Then, we +compute the building coverage growth rate over the 5-year +interval for the chosen cities using the model trained on +15,000 African images for 1200 epochs. +One challenge of temporal evaluation is the lack of build- +ing coverage ground truth data. To get a sense of how well +the proposed method tracks temporal changes in building +coverage, we use population change as a proxy for build- +ing coverage change. Specifically, we assume that popula- +tion and building coverage grows together, as more people +requires more buildings. However, we only use population +growth rate as a rough reference for the relative level of de- +velopments of different cities. +Results +Tracking changes in building coverage across +time allows us to understand the urban development of a +region, especially in cities or urban areas. We show that +the proposed model can be used to track building coverage +changes in regions of different levels of development and +provide the temporal experiment results in Figure 4. +4Sentinel-2 mission started in 2014. + diff --git a/29AzT4oBgHgl3EQffPxu/content/tmp_files/load_file.txt b/29AzT4oBgHgl3EQffPxu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08631be49c16d78eae032c8702357d21f54ba030 --- /dev/null +++ b/29AzT4oBgHgl3EQffPxu/content/tmp_files/load_file.txt @@ -0,0 +1,682 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf,len=681 +page_content='Building Coverage Estimation with Low-resolution Remote Sensing Imagery Enci Liu, Chenlin Meng, Matthew Kolodner, Eun Jee Sung, Sihang Chen Marshall Burke, David Lobell, Stefano Ermon Stanford University {chenlin, jesslec}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='edu, {mkolod, ejsung, schen22, mburke, dlobell}@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='edu, ermon@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='edu Abstract Building coverage statistics provide crucial insights into the urbanization, infrastructure, and poverty level of a region, fa- cilitating efforts towards alleviating poverty, building sustain- able cities, and allocating infrastructure investments and pub- lic service provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Global mapping of buildings has been made more efficient with the incorporation of deep learning models into the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' However, these models typically rely on high-resolution satellite imagery which are expensive to collect and infrequently updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As a result, building cov- erage data are not updated timely especially in developing re- gions where the built environment is changing quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In this paper, we propose a method for estimating building coverage using only publicly available low-resolution satellite imagery that is more frequently updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We show that having a multi- node quantile regression layer greatly improves the model’s spatial and temporal generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Our model achieves a co- efficient of determination (R2) as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='968 on predicting building coverage in regions of different levels of develop- ment around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We demonstrate that the proposed model accurately predicts the building coverage from raw in- put images and generalizes well to unseen countries and con- tinents, suggesting the possibility of estimating global build- ing coverage using only low-resolution remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Introduction The quantity and location of buildings provide important in- sight into the human activities and urban development of a region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Not only are building statistics themselves key socioeconomic indicators, they also help predict other key sustainable development indices, including poverty (Ayush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Uzkent, Yeh, and Ermon 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2020), population density (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2021), and climate out- comes (Chini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Moreover, building coverage statistics help policymakers and NGOs make informed deci- sions regarding the provision of public services, the targeting of humanitarian aid, and priorities for large-scale infrastruc- ture investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The development of deep learning detection (Redmon and Farhadi 2018) and segmentation models (Ronneberger, Fis- cher, and Brox 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2019) has allowed for more efficient global mapping of buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As a result, there has Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' been an increasing number of global settlement map datasets in the past decade (Esch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Marconcini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sirko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2021), allowing researchers to develop insights into the socioeconomic development of different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Nevertheless, detection- or segmentation-based methods typically rely on a large amount of high-resolution satel- lite imagery and the corresponding pixel- or instance-level labels for training, which are often prohibitively expensive and unaffordable for researchers and policymakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' More- over, high-resolution imagery are updated less frequently than low-resolution ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In addition, running detection or segmentation models over high-resolution images covering a large area requires a large amount of compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Due to these reasons, building data gathered in this way is often not updated timely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For example, the Microsoft Global Build- ing Footprints dataset1 is collected from satellite images be- tween 2014 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' However, during the 7-year period, population continues to grow and new buildings are con- structed, especially in fast developing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For instance, from 2015 to 2020, the population increased by 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='1% for Malappuram and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='2% for Abuja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='2 Moreover, fine- grained detection and segmentation models usually gener- alize poorly to unseen geographies, timestamps, and image sources because the appearance of buildings vary widely in satellite images (Yuan and Cheriyadat 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Compared with its high-resolution counterpart, low- resolution satellite imagery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sentinel-1 and Sentinel- 2) are publicly available and updated every month, making them desirable for studying the economic and urban devel- opment of a region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' However, prior works have not fully utilized low-resolution imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In this paper, we propose a cheaper and more generalizable way to update building statistics using only low-resolution satellite imagery from Sentinel-1 and Sentinel-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Instead of detecting or segment- ing each building in satellite images, the proposed model directly predicts the building coverage from the raw input pixels, using input imagery from a public source that is up- dated nearly weekly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, we found that incorporat- ing a multi-node quantile regression loss helps improve the generalization of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The proposed method achieves a coefficient of determination (R2) as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='968 in re- 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='com/microsoft/GlobalMLBuildingFootprints 2https://worldpopulationreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='com/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='01449v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='CV] 4 Jan 2023 gions from different continents and of different levels of de- velopment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We also conduct ablation studies and show that the incorporation of additional multi-spectral bands avail- able from the low-resolution satellite imagery and the multi- node quantile regression design help improve the model per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Method In this paper, we propose a deep-learning based regression model that accurately predicts building coverage in low- resolution satellite imagery from Sentinel-1 and Sentinel- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Unlike detection- or segmentation-based methods, our model does not require high-resolution training data or hand-crafted threshold and generalizes well to unseen re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The proposed method accurately predicts the building coverage in the raw input low-resolution satellite image with the help of a quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Problem Definition Given a geographical region, we want to estimate the build- ing coverage within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Due to the lack of direct statistics of building coverage in square meter or kilometer, we instead predict the number of building pixels y ∈ R, in the satellite image X representing the target region, assuming that the number of building pixels is proportional to the actual build- ing coverage (Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We want to build a model that predicts y from the raw image input X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Multi-node Quantile Regression Model We observe that the distribution of building pixel counts across Africa and South America are heavy-tailed, with more than 75% of the samples having less than 20% of all pixels being buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Regular linear regression trained on root-mean-square error objective estimates the conditional mean and assumes normality of the data distribution, fail- ing to model non-normal asymmetric distribution accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Moreover, linear regression is not robust to outlier values, which characterize the building pixel count distribution for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To address the aforementioned problems, we adopt quan- tile regression (Koenker and Bassett Jr 1978), which esti- mates the conditional quantile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=', median) of the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Quantile regression allows us to incorporate uncer- tainty in prediction and captures the relationship between the input and different quantiles of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As we will see in Section , multi-node quantile regression indeed empirically performed better than regular linear regression for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, we modify the ResNet18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2016) ar- chitecture to have K output channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' nodes), each cor- responding to a different quantile (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As it is ob- served that the median of a distribution gives the minimum absolute error from the ground truth (Hanley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2001), at inference time, we collect the model predictions from only the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5 quantile node, which is expected to predict the con- ditional median of the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Multi-node Quantile Loss For each node that represents a quantile q ∈ (0, 1), we com- pute an asymmetric quantile loss, or pinball loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Depending on the quantile q, over- and under-estimation are penalized unevenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, the node-wise pinball loss for a given prediction ˆy and ground truth label y is computed as follows: Lpinball(q, y, ˆy) = �q · |ˆy − y| if ˆy ≥ y (1 − q) · |ˆy − y| otherwise When q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5, the pinball loss is the same as the absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To compute the final loss, we take the mean of the pinball losses among the K output nodes as follows: Lquantile(y, ˆy) = 1 K K � n=1 Lpinball(qn, y, ˆy) where the subscript n indicates the index of the quantile in the list of quantiles predicted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Notice that when K = 1 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5, the quantile loss formula boils down to the regular L1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Experiment Setup Before introducing the experiment setups, we define tile and patch in the context of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We define a patch to be the small rasters of 50×50 pixels that we crop larger rasters into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' A tile is a larger satellite imagery that could be cropped into multiple smaller patches (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We train and eval- uate the multi-node quantile regression model on patches of 50 × 50 pixels and add post-processing steps to collect tile- level results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Training Data As the majority of the African continent is covered by for- est or desert and does not contain any buildings, we sample locations based on the population density so that the train- ing data contains sufficient tiles with buildings for the model to learn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Our training set contains 15,000 input-label pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sentinel-1 and Sentinel-2 images are used as as the input data and the Open Buildings dataset is used to derive the building coverage label, as we detail next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sentinel-1 (S1) satellites collect radar imagery for land and ocean monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We download the S1 satellite im- agery collected in 2020 from Google Earth Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The im- ages have 10m GSD and are composites that take the median value over the target period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We include band 1, the VV overall mean, as one of the input channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The S1 tiles are cropped into 50 × 50-pixel patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As areas with high built-up density are shown to result in stronger signals in S1 band 1 (Koppel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2017), we expect that adding this channel to the input will improve the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sentinel-2 (S2) satellites are equipped with the mission of land monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We download the 2020 composites of S2 imagery from Google Earth Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We include the RGB channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' bands 4, 3, 2) and the near-infrared (NIR) channel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' band 8) from S2 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Compared with other channels, NIR is useful for distinguishing the vegetation from the buildings (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Pessoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Schlosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' All of the four bands are available in 10m GSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The S2 tiles are cropped into 50 × 50-pixel patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' (a) Predict the number of building pixels 𝑦 in image 𝑋 𝑦 = 874 Base map (left) and the binary mask (right) used to derive the label 𝑦 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='9 Concat ResNet18 (b) Multi-node quantile regression model #𝑦 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5 ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' "#$%&\'( 𝐾 = 3 Input 𝑿 𝐴 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='96% Figure 1: (a) Given an input image X, we want to predict the number of building pixels y within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We assume that y ap- proximate the actual building coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' (b) Architecture of the multi-node quantile regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The input to the model includes five channels, which are Sentinel-1 band 1 (Overall Mean), Sentinel-2 band 4, 3, 2, 8 (R, G, B, NIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The model has K output nodes representing different quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Tile Patch Crop Summation 50 50 Figure 2: A tile can be cropped into multiple 50 × 50 pix- els patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The tile-level results are computed by taking summation of all the patch-level predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Open Buildings (Sirko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2021) contains 516M build- ing footprints across 43 African countries that cover 64% of the continent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The building footprints were detected us- ing state-of-the-art segmentation model collected from high- resolution imagery at different timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We downsample the original high-resolution mask (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5m GSD) to 10m GSD to match the resolution of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Then, we convert the continous-valued rasters into binary masks by threshold- ing at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The building pixel count labels are derived for the 50 × 50-pixel patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Experiment Settings To evaluate the generalization performance of the model, we consider three experiment settings that captures likely cases of real-world application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Holistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In the holistic setting, we train and test on data points sampled across the African continent based on popu- lation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Intra-country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In the intra-country setting, we train and test the model on samples from the same country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In the exclusive setting, we train the model on all African countries except for the one country on which we test our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Baselines To evaluate the performance of the proposed method, we use existing settlement map products from different years as baselines to compare against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' These off-the-shelf prod- ucts provide researchers and policymakers with general in- formation about urban shapes and boundaries, but could be less useful in providing up-to-date building coverage statis- tics, which change more frequently than the shape of urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We believe that these existing settlement map products serve as good baselines to compare our method against and provide information of them in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Gloal Urban Footprint (GUF) GUF was collected from 2011 to 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' It contains mappings of human settlements in the form of binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Global Human Settlement Layer (GHSL) GHSL was collected from Sentinel-1 images in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The building map is available as binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' World Settlement Footprint (WSF) WSF (Marconcini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2020) was collected in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' It contains binary masks of global human settlements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Evaluation Settings We evaluate the model performance at both patch-level and tile-level and describe the evaluation metrics in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Patch-level Evaluation As the model is trained on patches, we want to evaluate the model’s performance at the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We use two evaluation metrics for patch-level evaluation: mean absolute error (MAE) and Pearson’s r2 be- tween the predicted and the ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Tile-level Evaluation In real-world applications, building coverage statistics are needed over a large geography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To re- flect this use case, we also evaluate the model performance at 1661 (1687) 82 (83) 93 (87) 665 (645) 698 (704) 987 (945) Figure 3: Examples of model predictions in Brazil, which was unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The first row is the RGB input image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' the second row is the binary masks (where the bright yellow pixels are building pixels) from which we derive the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The model prediction is in black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' the ground truth labels are highlighted in green in the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Our method accurately estimated the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='/Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' settings* Open Buildings SpaceNet7 Holistic ✓ Intra-country ✓ Exclusive ✓ Patch-level ✓ ✓ Tile-level ✓ Table 1: Checkmarks indicate that the corresponding dataset is used as validation data under the experiment (Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=') or evaluation (Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=') settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' *Different experiment settings require retraining the model, while different evaluation set- tings do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' tile-level using absolute error in building coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To com- pare the statistics of building coverage across baselines with different GSDs, we compute the percentage of building pix- els within a tile as a proxy for building coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For a tile R cropped into N patches at inference time, let yi be the number of building pixels in patch i, the building coverage percentage is computed as: Cbuilding(R) = �N i=1 yi Htile × Wtile × 100 where Htile and Wtile denote the height and width (in pixel) of the tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The absolute error between a method and the ground truth is computed as the absolute difference between the two building coverage percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Evaluation Data We evaluate our model on Open Buildings and SpaceNet 7 Challenge datasets and provide the information in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Open Buildings The Open Buildings (Sirko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2021) dataset provides the training labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We evaluate the model performance on a hold-out test subset of the Open Buildings dataset at patch-level only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We do not use Open Buildings for tile-level evaluation because it contains data from different timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' SpaceNet 7 Challenge dataset (SpaceNet7) SpaceNet73 was published in 2020 as the data for the SpaceNet 7 Multi- Temporal Urban Development Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The dataset pro- vides 4km × 4km tiles and the building polygons in each tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We downsample the raster to 10m GSD to match the resolution of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We evaluate the model perfor- mance on SpaceNet7 at both patch-level and tile-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We use the SpaceNet7 tiles for validation because the la- bels were collected the same year as the Sentinel-1/-2 in- put data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Furthermore, SpaceNet7 includes tiles from re- gions outside of Africa, allowing us to evaluate the model’s performance on unseen countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, we evaluate our model on six SpaceNet7 tiles from different regions: Uganda, Zambia, Ghana, Peru, Brazil, and Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Note that we do not use SpaceNet7 as the training labels be- cause the data is limited in quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' These labels are human- generated and likely of higher quality compared to those from Open Buildings, which are generated by a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Results In this section, we evaluate the performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We first show that our model accurately predicts building coverage in Africa and generalizes to South Ameri- can regions unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We also conduct ablation studies demonstrating that the major design choices – non- RGB bands and multi-node quantile regression – are neces- sary for boosting model performance and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 3SpaceNet on Amazon Web Services (AWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' “Datasets.” The SpaceNet Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Last modified October 1st, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Accessed on November 20th, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' https://spacenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='ai/datasets/ Africa South America Uganda Zambia Ghana Brazil Peru Mexico Method R2 ↑ Tile ↓ R2 ↑ Tile ↓ R2 ↑ Tile ↓ R2 ↑ Tile ↓ R2 ↑ Tile ↓ R2 ↑ Tile ↓ GUF (2012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='092 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='466 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='59 – – – – – – WSF (2015) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='286 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='36 Table 2: Results on SpaceNet7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The bottom four rows are the proposed methods with the corresponding components removed and the complete version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In the table, r2 is the patch-level Pearson’s r2 and Tile is the tile-level absolute error between SpaceNet7 and the corresponding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The model was trained with 15,000 samples in Africa for 1200 epochs and tested on the corresponding SpaceNet7 tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' setting Train Test MAE ↓ R2 ↑ Holistic Africa Africa 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='888 Intra-country Rwanda Rwanda 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='938 Exclusive Africa* Rwanda 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='844 Exclusive Africa* Uganda 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='568 Excluisve Africa* Kenya 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='915 Table 3: Patch-level results for different experiment settings on Open Buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' All models in the table are trained with 15,000 samples from the corresponding train regions for 1200 epochs and tested on 1,000 samples from the corre- sponding test regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' *Under the Exclusive setting, the cor- responding test region is removed from the training set so that the model is tested on unseen regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Accurate Prediction in Africa To demonstrate the effectiveness of our method, we evaluate it on both SpaceNet7 (see Table 2) and Open Buildings (see Table 3) in Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We define a tile as multiple patches and compute the tile-level results by taking summation of the patches within it (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As shown in Table 2, our model achieves an R2 as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='968 at patch-level, outperforming the baselines on most of the African regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Some examples of the proposed model’s patch-level prediction are provided in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Furthermore, the proposed method yields fairly accurate building cov- erage estimates at tile-level, achieving a low error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='54% on Ghana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Additionally, we observe that baselines like WSF and GUF give good estimates at only tile-level but not the other, indicating that the errors at patch-level are cancelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In contrast, the proposed method yields con- sistently accurate estimations at both tile- and patch-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We emphasize that having good performance on both scales is ideal, since it gets us closer to finer-grained pixel-level predictions (semantic segmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Generalization to Unseen Regions Prior methods for generating building coverage statistics generalize poorly under domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To evaluate the gen- eralizability of the proposed method, we train and test the multi-node quantile regression model under three exper- iment settings (defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='3) – Holistic, Intra- country, and Exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We provide the patch-level results on the Open Buildings dataset in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We see that among the three experiment settings, Intra- country gives the highest Pearson’s r2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='971 because the model is tested on in-domain data and the region is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We also observe that the proposed multi-node quantile re- gression model generalizes well to regions not seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Speicifcally, under the Exclusive setting, the pro- posed method achives an r2 as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='962 even with the test regions removed from the training set (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Furthermore, the proposed model generalizes to regions outside of the African continent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We evaluate our method on SpaceNet7 tiles from South America and provide the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We observe that the proposed model achieves comparable or superior performance when evalu- ated on Brazil, Peru, and Mexico compared with the base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' This generalization to a different continent indicates that our model could potentially be applied to the globe for collecting building coverage statistics, while using only pub- licly available low-resolution satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Ablation Studies In this part, we carry out ablation studies on 1) the incorpo- ration of S1 band 1 and S2 band 8 as input and 2) the multi- node quantile regression, and provide the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Multi-node Quantile Regression To see the effect of hav- ing multiple output nodes for different quantiles, we com- pare the performance of the multi-node quantile regression model with the single-node model trained on the L1 objec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We provide the ablation study results on SpaceNet7 in Table 2 (see row “w/o multi-node QR”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We observe that the proposed multi-node model outperforms the single-node model by a large margin on all test regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In addition, as shown in the scatter plots for ground truth VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' predicted building pixel counts (Figure 4), the model tends to overesti- mate the building coverage without the multi-node quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We speculate that having multiple output nodes improve performance because the pinball losses from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='9 quantile nodes help bound the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Multi-Spectral Bands We conduct ablation studies on the two non-RGB multi-spectral bands — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' S1 band 1 and S2 w/o S2 band 8 w/o S1 band 1 w/o Multi-node QR Ours 𝑹𝟐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='751 𝑹𝟐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='864 𝑹𝟐 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='069 𝑹𝟐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='968 Ground truth building pixel count Predicted building pixel count Figure 4: Scatter plots of patch-level predicted number of building pixels against the ground truth from the ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' All models are trained on Africa and tested in the Brazil tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Removing the multi-node quantile regression or any of the non-RGB bands makes the model be prone to overestimation or underestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' band 8 — and provide the ablation study results evaluated on SpaceNet7 in Table 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We observe that at both patch- and tile-level, incorporating S1 band 1 and S2 band 8 boosts the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' From the scatter plots in Figure 4, we see that removing either S1 band 1 or S2 band 8 makes the model un- derestimates at patch-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' This suggests that the non-RGB bands provide additional information that helps correct the model’s tendency to under- or over-estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Discussion and Social Impact United Nations’ Sustainable Development Goals (SDGs) present an urgent call for action in all countries and collabo- ration between different sectors of policy-making for a more sustainable development (Nations 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' However, relevant data for informing the decision makers and organizations are often lacking or infrequently collected, especially in fast de- veloping countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Building coverage is an important so- cioeconomic indicator and also helps predict other indica- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In this paper, we develop a framework for estimat- ing building coverage using only free low-resolution satel- lite imagery from Sentinel-1 and Sentinel-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The proposed multi-node quantile regression model yields fairly accurate estimates and generalizes well to unseen countries and con- tinents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' This paper offers a cost-efficient and generalizable way to collect global building coverage statistics, accelerating the progress towards multiple SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For example, build- ing coverage helps predict the level of economic develop- ment, which informs policymakers’ decisions for alleviating poverty (SDG 1 No Poverty), allocating infrastructure re- sources (SDG 9 Industry, Innovation and Infrastructure), and building more sustainable cities (SDG 11 Sustainable Cities and Communities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In addition, building coverage provides important information about the interaction between human and environment, including the monitoring of agriculture to reduce hunger (SDG 2 Zero Hunger) and climate measure- ment (SDG 13 Climate Action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Furthermore, the proposed method could potentially be applied to track changes of building coverage for different regions in the world, assisting existing efforts for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For example, United Nations’ World Urbanization Prospects re- port provides estimates and projections of urban and rural data, including population and area, throughout the time (Nations 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The proposed method could be applied to derive building coverage statistics as soon as satellite images are updated (the low-resolution satellite imagery are updated on a weekly basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' As building coverage is highly corre- lated with or can be used to derive other values like building density, urban area, and population, our method could po- tentially aid the efforts to track urban development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' This paper demonstrates the viability of using low- resolution free satellite imagery for estimating building cov- erage statistics over a large geography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Future research could 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Perez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Driscoll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Azzari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Tang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Lobell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Ermon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' and Burke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Using publicly available satellite imagery and deep learning to understand economic well-being in Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Nature Communications, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' and Cheriyadat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Learning to count buildings in diverse aerial scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' In Proceedings of the 22nd ACM SIGSPATIAL International Conference on Ad- vances in Geographic Information Systems, 271–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Appendix A: Datasets Sentinel-1 provides free global Synthetic Aperture Radar (SAR) imagery available with 9 bands at ground sampling distance (GSD) of 10m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The revisit time of the Sentinel-1 satellite is 12 days, meaning that the images are updated fre- quently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We include band 1, the VV overall mean, as one of the input channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Each Sentinel-1 tile that we download is 2km-by-2km and has the shape 200×200 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Each tile is then cropped into 16 smaller patches of shape 50×50 pixels when fed into our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Sentinel-2 provides multi-spectral images in from the vis- ible to the shortwave infrared spectral range (SWIR) with the revisit time of 10 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The Sentinel-2 satellite imagery con- tains 13 multi-spectral bands with GSDs ranging from 10 to 60m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Besides the RGB channels, we also included the near- infrared (NIR) channel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' band 8) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Like Sentinel- 1, all the Sentinel-2 imagery are downloaded as 2km-by- 2km tiles and cropped into 16 patches of size 50 × 50-pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Open Buildings contains 516M building footprints across 43 African countries that cover 64% of the continent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The building footprints were detected using state-of-the-art seg- mentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The Open Buildings data is originally in the format of polygons labeled with three confidence inter- vals of buildings presence - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='65 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='7, and greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For our purpose, we download the Open Buildings data as high-resolution rasters with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5m GSD and two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Band 1 demonstrates the model confidence that a building is located in the region with the confidence interval preserved from the original polygon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' A band 1 value of zero indicates no building presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Band 2 is a reclassifica- tion of the confidence scores into four buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We use band 1 to derive the binary label of building and non-building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, we first downsample the rasters to 10m GSD to match the resolution of the Sentinel-1 and Sentinel-2 input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Then, we convert the continous- valued band 1 into a binary mask by treating all pixels with a non-zero value as a building pixel – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' a pixel that contains buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Like Sentinel-1 and Sentinel-2, the downsampled binary mask is cropped into 50 × 50-pixel patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For each smaller patch, we use the number of building pixels as the labels for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Appendix B: Baselines Figure 5 shows the visualization of different benchmarks de- scribed below and datasets we used in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Gloal Urban Footprint (GUF) is a worldwide mapping of human settlement patterns in the form of binary masks available in 12m GSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The building footprints are derived from satellite images of TerraSAR-X and TanDEM-X from 2011 to 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Global Human Settlement Layer (GHSL) was collected from backscattered information of Sentinel-1 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The building map is available as binary masks at 20m GSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' World Settlement Footprint (WSF) is a binary mask of global human settlements available in 10m GSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The map Base map Open Buildings (2021) WSF (2015) GHSL (2018) GUF (2012) SpaceNet (2020) Figure 5: Binary settlement maps in Zambia (1 = red build- ing pixel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 0 = white non-building pixel) from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The years of collection for the settlement maps are provided in the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' (a) Geo-locations of training samples (b) Per km2 cost and resolution of different satellite imagery Figure 6: (a) The geo-locations of our training samples, all of which are from Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' (b) High-resolution satellite im- agery are expensive, while many lower-resolution ones are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' was derived from Sentinel-1 and Landsat-8 satellite imagery in year 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Appendix C: Experiments Model Implementation We modify the ResNet18 architecture to incorporate multi- node quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, we replace the final fully-connected (FC) layer in ResNet18 with two FC layers, each followed by a ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We set the number of output channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' nodes) to be K, which is the number of quantiles the model predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For all models in this pa- per, we use K = 3 and predict the quantiles {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Each channel corresponds to a quantile and the correspond- ing pinball loss is computed using the output of that partic- ular channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To prevent the model from overfitting, we add dropout layers after each ResNet block, which we found em- pirically that it helps the model generalize to unseen regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Price($/km^2) GSDmRegion Population (%) Building (%) Dhaka 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='23 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='68 Kampala 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='28 Athens 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='55 Boston 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='45 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='42 Table 4: Temporal change experiment results on four cities of different levels of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The second and third are percentage change from 2016 to 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' All results are col- lected from the model trained with 15,000 samples in Africa for 1200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Training Details The training samples’ geo-locations are shown in Figure 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' For all models in the experiment section, we train them on 15000 samples for 1200 epochs with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' The models have fully converged at the point where we end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Temporal Experiments Evaluation To evaluate our model’s ability to track tem- poral changes in building coverage, we run the model on satellite images from 2016 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content='4 Specifically, we chose four cities with various levels of development from four dif- ferent continents – Dhaka, Kampala, Athens, and Boston – and downloaded all images covering the cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Then, we compute the building coverage growth rate over the 5-year interval for the chosen cities using the model trained on 15,000 African images for 1200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' One challenge of temporal evaluation is the lack of build- ing coverage ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' To get a sense of how well the proposed method tracks temporal changes in building coverage, we use population change as a proxy for build- ing coverage change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Specifically, we assume that popula- tion and building coverage grows together, as more people requires more buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' However, we only use population growth rate as a rough reference for the relative level of de- velopments of different cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' Results Tracking changes in building coverage across time allows us to understand the urban development of a region, especially in cities or urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' We show that the proposed model can be used to track building coverage changes in regions of different levels of development and provide the temporal experiment results in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} +page_content=' 4Sentinel-2 mission started in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQffPxu/content/2301.01449v1.pdf'} diff --git a/59AzT4oBgHgl3EQfEfpg/content/tmp_files/2301.00994v1.pdf.txt b/59AzT4oBgHgl3EQfEfpg/content/tmp_files/2301.00994v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..81bc5c5d5910048ac2fc1d08c98b3cf29a168396 --- /dev/null +++ b/59AzT4oBgHgl3EQfEfpg/content/tmp_files/2301.00994v1.pdf.txt @@ -0,0 +1,784 @@ +arXiv:2301.00994v1 [quant-ph] 3 Jan 2023 +Pinhole quantum ghost imaging +Andres Vega,1, a) Sina Saravi,1 Thomas Pertsch,1, 2 and Frank Setzpfandt1 +1)Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena, Albert-Einstein-Str. 15, 07745 Jena, +Germany +2)Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Str. 7, 07745 Jena, +Germany +(Dated: 4 January 2023) +We propose a quantum ghost imaging scheme based on biphotons, that, by using a collimated pump beam of the right +size for biphoton generation, obviates the need for lenses to achieve imaging. The scheme is found to be analogous +to the classical pinhole camera, where we show that the equivalent to the classical pinhole size depends mainly on the +width of the pump beam, but also on the thickness of the nonlinear crystal and the wavelengths of the biphoton. +Quantum ghost diffraction1 and imaging2 rely on the spa- +tial correlations of a biphoton, which can be created by the +nonlinear process of spontaneous parametric down conversion +(SPDC)3,4, where a pump (P) photon impinging on a crystal +with second-order nonlinearity χ(2) is split into a pair of pho- +tons called signal (S) and idler (I). After creation, the two +photons are separated into two different paths and only one of +them interacts with the object, e.g. the signal photon. Then, +the signal photon is measured by a detector with no spatial res- +olution, whereas another detector with spatial resolution mea- +sures the idler photon that never interacted with the object. +None of the detectors alone can recover a diffraction pattern +or image of the object. Remarkably, these can be retrieved +by correlating the two measurements5,6. This measurement +technique has two main advantages. First, very low numbers +of photons can be used due to the inherently better signal-to- +noise ratio of quantum ghost imaging compared to imaging +with classical light7,8. Additionally, ghost imaging with two- +color biphotons can overcome limitations due to inaccessible +wavelength ranges for illumination and detection9–11. +To form a ghost image, usually lenses are placed in the path +of the signal and/or idler after the crystal2,7,8 or in the pump +beam before the crystal12. The lenses introduce a parabolic +phase-front in either of the beam paths, which results in +the formation of the image in the coincidence measurement. +However, quantum ghost imaging can be also realized without +lenses by adding the parabolic phase-front through engineer- +ing the nonlinearity profile of the nonlinear crystal, e.g. by +using a nonlinear photonic crystal13. Furthermore, as ghost +imaging can be also realized with classical light using ther- +mal light sources14–19, their inherent property of acting like +a phase-conjugated mirror16 in a ghost imaging scheme can +also be used for lensless ghost imaging20,21. +In classical optics, lensless imaging can be also realized us- +ing the principle of pinhole imaging22,23. In a pinhole camera, +the object is located on one side of an opaque screen with a +small pinhole, whereas the detector is on the other side. With- +out the need of lenses, the detector captures a shadow of the +object, which can be optimized by adapting the pinhole size22. +Throughout this manuscript, we will refer to this shadow as an +image although strictly no imaging is taking place. For appli- +a)Electronic mail: andres.vega@uni-jena.de +cations where high spatial resolution is not needed, this type +of lensless imaging has several advantages over imaging with +lenses, among which are a larger depth of field, a wide angular +field of view23, and its applicability in wavelength ranges for +which high-quality lenses are less available24,25. +In this work, we want to show that the advantages of pin- +hole imaging can be also harnessed in quantum ghost imag- +ing. For ghost imaging with thermal light, a pinhole-based +scheme has been already proposed, where the optimal lensless +imaging condition depends on the size of the thermal source26. +We extend this approach of lensless imaging to the quantum +regime with entangled photons based on the setup sketched +in Fig. 1. Here, we assume that the nonlinear crystal gener- +ating photon pairs is illuminated by a collimated pump beam +and, contrary to ghost imaging with thermal light, we use a +bucket detector instead of a point detector behind the object. +We will show, that for specific pump beam diameters, pinhole +quantum ghost imaging can be realized and we investigate its +properties and optimum regime of operation. To this end, we +start by discussing the biphoton joint spatial probability (JSP) +and the quantum ghost pattern (G) together with a numeri- +cal example. Later, we derive a simplified analytical model +for our imaging scheme that explains the observations of the +numerical example and allows the connection to the classical +pinhole camera. Using this model, we will furthermore dis- +cuss the spatial resolution of pinhole quantum ghost imaging. +Throughout this work, z is the propagation direction and +we restrict our analysis, without loss of generality, to one +transverse dimension x in position space, whose conjugate in +momentum space is the transverse component of the wave- +vector kx. We also assume an infinitely extended nonlinear +crystal in the transverse direction, which ensures transverse +Nonlinear +crystal +Bucket +detector +Object +Coincidence +circuit +Spatially +resolving detector +FIG. 1. Sketch of the considered setup. + +2 +phase matching kxP = kxS + kxI. We assume an undepleted +monochromatic classical pump beam of the form EP(x,z,t) = +� dkxP φP(kxP) exp[i(kxPx+ kzPz− ωPt)], where φP(kxP) is the +pump spatial spectrum, the longitudinal component of the +wave-vector is kz = [(ωn/c)2 − k2 +x]1/2 with ω/c = 2π/λ, λ +being the wavelength in free space and ω the corresponding +frequency. Additionally, we ignore the effects of the bound- +aries between the crystal and its surrounding free space, which +would cause refracted and reflected waves, and assume a sim- +plified case with the crystal’s refractive index n = 1. We in- +vestigate signal and idler photons at fixed frequencies of ωI +and ωS, such that ωP = ωS + ωI. This can be achieved exper- +imentally by placing narrow bandpass filters centered around +these frequencies in their beam paths. Under these condi- +tions, the biphoton quantum state after the filters will have +the form27 |Ψ⟩ ∝ +� dkxSdkxI ψSPDC(kxS,kxI) |kxS,ωS⟩|kxI,ωI⟩, +with ψSPDC(kxS,kxI) = φP(kxS + kxI) sinc(∆kz lz/2). +Here, +∆kz = kzP − kzS − kzI, lz is the thickness of the crystal, and +|kx,ω⟩ is the single-photon state defined by the transverse +component of the wave-vector kx and frequency ω. +The JSP of the biphoton state at the two detectors in Fig. 1 +is28 +JSP(xS,xI) ∝ +���F−1� +hI(kxI)h2S(kxS) +� +to(kxS)∗ +� +h1S(kxS) +×ψSPDC(kxS,kxI) +������ +2 +, +(1) +where F−1 is a two-dimensional inverse Fourier transform, +(kxS,kxI) → (xS,xI), and h are free space transfer func- +tions with hI(kxI) = exp(ikzI zI), h1S(kxS) = exp(ikzS d) and +h2S(kxS) = exp[ikzS(zS − d)]. Finally, ∗ denotes the convo- +lution only in kxS as the object with transmission To(xS) is +in the signal arm. +The ghost pattern G(xI) for a bucket +detector that collects all signal photons is derived from the +JSP as G(xI) ∝ +� dxS JSP(xS,xI). We assume a pump beam +with Gaussian spatial spectrum, whose waist is located at +the center of the nonlinear crystal at the plane z = 0, where +φP ∝ exp[−σ2 +P(kxS + kxI)2/2] has a flat wave front and a width +σP in position space (see zoomed out region in Fig. 1). +As shown in pinhole ghost imaging with a thermal source26, +the size of the photon source has a similar role as the pin- +hole size in classical optics, determining the optimal regime +of imaging. This suggests that in the quantum regime, the +same role can exist for the size of the biphoton source, which +depends on the width of the pump beam. To examine this +premise, we numerically calculate quantum ghost imaging in +a setup with a nonlinear crystal with thickness lz = 3 mm, +a pump with wavelength of λP = 350 nm, degenerate down- +converted photons with λS = λI = 700 nm, and detectors lo- +cated at zS = 1.2 m and zI = 1.5 m from the crystal. As ob- +ject, we consider a double-slit with 940 µm slit separation, +with unity transmission in each slit of 50 µm width and no +transmission elsewhere. +In Fig. 2(a-d) we present the en- +suing normalized JSPs calculated using Eq. (1) for different +sizes of the pump beam. The double-slit always results in +two spots, whose separation is approximately five times larger +than the slit separation. Depending on the width of the pump +beam, they change their widths and begin to overlap and in- +terfere. The interference is minimal in Fig. 2(c) with pump +0 +1 +(d) +(e) +(a) +(c) +-8 +0 +8 +-8 +0 +8 +-8 +0 +8 +-8 +0 +8 +-8 +0 +8 +102 +103 +(b) +(c) +0 +1 +(d) +(a) +(b) +FIG. 2. (a-d) JSP(xS,xI) and corresponding (e) quantum ghost pat- +tern G(xI) of a double-slit, 940 µm slit separation and 50 µm slit +width, located at d = 30 cm produced by a pump width σP of (a) +58 µm, (b) 102 µm, (c) 167 µm, and (d) 800 µm. +width σP = 167 µm. In Fig. 2(e), we show the corresponding +ghost patterns, where the cases of (a-d) are marked. We see, +that for a specific range of pump widths, two separate maxima +are visible, corresponding to an image of the double-slit. We +note, that the well-known quantum ghost diffraction pattern1 +of the object could be recovered using a large pump width, +as in Fig. 2(d), and replacing the bucket with a point detector +which would measure only a horizontal cut through the JSP. +This numerical example portrays the core idea of this work. +Other schemes12,13 have shown that a pump wave with a +curved wave front, obtained by means of a lens placed be- +fore the nonlinear crystal or using a photonic crystal, can also +be used for quantum ghost imaging without lenses in the paths +of the biphoton. Fig. 2(c) now shows, that lensless quantum +ghost imaging can be also achieved by simply using a colli- +mated pump beam with an optimal width. This is easily seen, +as the Rayleigh length29 of the pump, 2πσ2 +P/λP = 50 cm, is +much larger than the crystal thickness, lz = 3 mm. +To find the optimal conditions for this imaging scheme, +we derive a simplified analytical model. Here, we consider +only one of the slits and assume it has infinitesimal width and +is located at xS = a, which means that To(xS) = δ(xS − a). +This object is put into Eq. (1) and in paraxial approxima- +tion an analytical solution can be calculated (see supplemen- +tary material), where we approximate the sinc function ap- +pearing due to phasematching in the nonlinear crystal by +a Gaussian30. +We find a Gaussian ghost intensity pattern +G(xI) ∝ exp[−(xI − x0)2/(2σ2 +G)] with a width σG and a max- +imum located at x0, given by +σG = +� +2Re +� +α−1 +1 +��−1/2, +(2) +x0 = a Re +� +α−1 +1 α2 +�� +Re +� +α−1 +1 +��−1 , +(3) +where +α1 = σ2 +P + γ lz +π (λI − λP)+ iλIzI +2π − +� +σ2 +P − γ lzλP +π +� +α2, +(4) +α2 = +� +σ2 +P − γ lzλP +π +�� +σ2 +P + γ lz +π (λS − λP)+ idλS +2π +�−1 +(5) + +102 +-8 +0 +83 +with γ = 0.455/4, a constant that comes from the sinc to Gaus- +sian approximation. Due to the bucket detector that collects +all signal photons behind the object, the equations do not de- +pend on zS, the distance of the object to the bucket detector; +however, the model does depend on the location of the re- +solving detector zI. These distances remain at zS = 1.2 m and +zI = 1.5 m throughout the manuscript. Fig. 3 shows the width +σG of the ghost pattern and the normalized position x0/a with +respect to the width of the pump σP and the distance of the ob- +ject to the crystal d. We observe in Fig. 3 that, for each object +position d, the width of the ghost pattern σG has a minimum +at a certain pump width σP. This confirms the observations of +the numerical example in Fig. 2 that uses d = 30 cm, where +the optimal case for imaging, marked with a dot in Fig. 2(c), +leads to the narrowest ghost pattern for each of the slits. In +the rest of the cases, the pattern of each slit is too wide, result- +ing in considerable overlap between them and hence the loss +of visibility of the ghost pattern. If a second infinitesimal slit +would by at xS = −a, the distance between the two maxima in +the ghost patterns would be 2x0, therefore, x0/a represents the +magnification. Fig. 3(b) verifies that for the numerical exam- +ple in Fig. 2 the magnification is approximately equal to five. +It also tells that the magnification is always negative, implying +that the ghost image is inverted. +Fig. 3(a) shows, that the minimum width of the ghost pat- +tern becomes smaller as the object is placed farther from the +crystal. This value, however, does not reach zero, the behavior +of σG converges to approximately the orange curve in the limit +where the object is very distant from the crystal, d → ∞. Here, +Eq. (2) can be reduced to σ2 +G = σ2 +0 + σ−2 +0 +[zIλI/(4π)]2. The +width of the ghost pattern of an infinitesimal slit object with +the spatially resolving detector placed right after the crystal, +σ0 = σG(zI = 0), is +σ0 = +�1 +2σ2 +P + γ +� λI +λS +��λPlz +2π +��1/2 +, +(6) +an expression that depends only on the parameters of the +biphoton source. For a fixed value of zIλI, the ghost pattern +width σG has a minimum value, namely σmin +G += +√ +2σ0, when +σ2 +0 = zIλI/(4π). Remarkably, this result is equivalent to the +optimal pinhole size σpinhole in a classical pinhole camera that +creates the smallest point image of a slit upon spatially inco- +herent illumination22, σ2 +pinhole ∝ λz. Hence, σ0 can be consid- +ered the pinhole size of quantum ghost imaging. It does not +only depend on the width of the pump but also on the thickness +of the crystal and the biphoton wavelengths. However, the de- +viation of the equivalent pinhole size σ0 from the pump width +σP due to the biphoton wavelength is small as for a pump with +negligible diffraction inside the crystal σ2 +P ≫ λPlz/(2π). +Next, we analyze the magnification to complement the anal- +ogy. +The magnification of the classical pinhole camera is +given by geometric optics as −z/d with z the distance of the +detector to the pinhole and d the distance of the object to the +pinhole. A similar relation can be found from the analytical +model of the proposed ghost imaging scheme using Eq. (3), +under the conditions that σ2 +P ≫ (γ lz/π)(max(λS,λI) − λP) +and {dλS/(2π), zIλI/(2π)} ≫ σ2 +P. The first condition en- +sures, that signal, idler and pump waves have negligible +(a) +(b) +0.4 +1 +1.6 +2.2 +2.8 +0 +200 +400 +-45 +-30 +-15 +0 +0 +200 +400 +d=3cm +↓ +d=5cm +d=10cm +d=30cm +d→∞ +FIG. 3. (a) Width σG and (b) magnification of the position x0/a of +the ghost pattern of a infinitesimal slit at xS = a with respect to the +pump width σP at various distances d of the slit to the crystal. The +circle marks the parameters of the numerical example in Fig. 2(c). +diffraction inside the nonlinear crystal. In this case, σP de- +fines the size of the generated signal and idler beams inside the +crystal, and hence their Rayleigh lengths are also much larger +than the crystal thickness lz. The second condition states that +both the object and the resolving detector are in the far-field +of the biphoton source. Under these conditions, following the +steps detailed in the supplementary material, we find the mag- +nification to be x0/a ≈ −(zI/d)(λI/λS). This is similar to +the geometrical magnification of the classical pinhole camera +but including again the biphoton wavelengths. From this ex- +pression, the magnification of the numerical example in Fig. 2 +can be quickly found −(1.5m/30cm)(700nm/700nm) = −5, +see the supplementary material for an example with non- +degenerate wavelengths. Noteworthy, our imaging scheme is +not limited to the far-field domain. We only take the approxi- +mations to find the simple expression for the magnification to +build the analogy with the classical pinhole camera. +We further harness the analytical model of Eqs. (2) and (3) +to derive the transverse resolution of pinhole quantum imag- +ing. In the following, we use Rayleigh’s resolution criterion +that is defined as the minimum distance between two point- +like objects to distinguish them from one another29. A smaller +ghost pattern width σG of a point-like object results in a bet- +ter distinction between neighboring objects, as was demon- +strated in Fig. 2. At the same time, a larger magnification +|x0/a| would results in a better distinction between two neigh- +boring objects, as their images are further apart. This suggests +that, to optimally tell two objects apart, not only the ghost pat- +tern width has to be taken into account but also the magnifi- +cation. To test this hypothesis, we begin by taking the derived +Gaussian ghost pattern from an infinitesimal slit A at xS = a. +For the sake of simplicity we normalize its maximum to one, +resulting in the ghost pattern GA = exp[−(xI − x0)2/(2σ2 +G)], +where σG is given by Eq. (2) and x0 by Eq. (3). If we in- +clude another similar slit B but at xS = −a, the resulting ghost +pattern is symmetric with respect to xI = 0 and is given ap- +proximately by G ≈ GA + GB, assuming negligible interfer- +ence. The two slits can be distinguished in this pattern when +its visibility is above a certain threshold, here heuristically +chosen to be 0.4. +That means, the intensity at xI = 0 be- +tween the two maxima should be smaller than 0.4 of the maxi- +mal intensity at xI = ±x0, which gives the threshold condition +G(xI = 0)th ≈ GA(0)+GB(0) = 2GA(0) = 0.4. Using this, the + +4 +0 +100 +200 +300 +1.3 +1.6 +0.1 +0.4 +0.7 +1 +d=3cm +d=5cm +d=10cm +d=30cm +� +0 +1 +0 +1 +-10 +0 +10 +(�� +�� � +��� +� + +d=1m +100 +101 +102 +10-1 +100 +101 +d + +∞ +d=3cm +d=30cm +d=1m +FIG. 4. (a) Resolution R with respect to the pump width σP. The +green line connects the minima of curves of a wider range of object +distances d. (c) Number of spatial modes N with respect to the crystal +thickness lz at various d. Numerically, JSP (top) and G (bottom) of +a (b) double-slit and a (d) complex object (magnified transmission in +yellow). Circles in (a) and (c) mark the parameters of (b) and (d). +expression for GA, and Eqs. (2) and (3), we find the transverse +spatial resolution R corresponding to the minimum resolvable +distance 2a between two identical infinitesimal slits to be +R = 2 +� +−ln[G(0)th/2]Re +� +α−1 +1 +��1/2���Re +� +α−1 +1 α2 +���� +−1 +. +(7) +Fig. 4(a) displays its dependence on the pump width σP at dif- +ferent object distances d. The thick green line connects the +minima of several curves over a wider range of d to portray +the tendency. One could naively expect from the width of +the ghost pattern σG in Fig. 3(a) that the resolution R could +improve as the object is farther from the crystal since the min- +imum width becomes smaller. However, Fig. 4(a) tells the +opposite, as R is enhanced by the large magnification at small +object distances d. This is confirmed numerically by consid- +ering again the case depicted in Fig. 2(b), that uses a pump +width of 102 µm and d = 30cm, and changing the position of +the object to d = 10cm. The resulting JSP and ghost pattern G +are shown in Fig. 4(b). Compared to Fig. 2(b), the visibility is +increased due to the larger magnification. This originates from +the fact that, for a fixed object distance d, the minimum ghost +pattern width σG does not coincide with the largest magnifi- +cation x0/a. Hence, the optimal pump width σP that results in +the best resolution is not necessarily when σG is minimized. +This is only true for larger values of d where x0/a is almost +independent of σP, as in the numerical example of Fig. 2. +Noteworthy, the green tendency line in Fig. 4(a) hints that +the closer the object is to the crystal and the smaller the pump +width, the better the resolution R. However, the smaller the +pump width, the broader its spatial spectrum becomes, in such +case the used paraxial approximation does not hold. A non- +paraxial formulation is a matter of future research. Addition- +ally, R will depend linearly on the object position d for large +values of d. This is because in such case, the ghost pattern +width σG is nearly independent of d, see Fig. 3(a), and the +magnification is |x0/a| ∝ 1/d, as already explained. +In addition to the resolution, it is also of great interest to +describe the extent of the signal illumination onto the object, +which limits the object size that can be imaged. This is finite +and can be found from a projection of the JSP at the object po- +sition onto the signal axis (see the analytical expression in the +supplementary material). This illumination size σS increases +with the position of the object d and decreases with the crystal +thickness lz. Importantly, the ratio between the illumination +size and the resolution tells the maximum number of identical +infinitesimal slits that can be resolved inside the illuminated +region of the object, N ≡ σS/R, i.e. describes the number of +independent spatial modes in the object illumination31. Its de- +pendence on the crystal thickness lz is displayed in Fig. 4(c) +for various d, each at a pump width σP that minimizes the res- +olution as described by the green line in Fig. 4(a). To increase +the number of spatial modes N, a thinner crystal can be used, +similar to other quantum imaging schemes2,32, as it allows a +larger range of transverse wave-vectors33. Also, the object +could be put farther away from the crystal. Here, the increase +of N with d has a limit due to the linear dependence of both +σS and R on d for very distant objects. Finally, we found that +photon-pairs with non-degenerate wavelengths show a minor +improvement in the resolution and number of modes, see the +supplementary material for some examples. +Lastly, we sum up the main features of the proposed setup +with a ghost image of a more complicated object with vari- +ous shapes and transmissions, see Fig. 4(d). This object is +placed at d = 1 m and we use a pump width σP = 258 µm that +optimizes the ghost image resolution to R = 1.5 mm, allows +N = 10 spatial modes and has a magnification x0/a = −1.2. +Moreover, unlike the setup with a pseudo-thermal source26, +we see in the JSP of this example that an integrating bucket +detector is necessary to show the whole illuminated section of +the object in the ghost pattern, a point detector would not be +able to detect signals from all objects as it would just “take a +horizontal thin slice” of the JSP. +To conclude, we proposed a quantum ghost imaging +scheme without lenses in the biphoton arms by means of a +collimated pump beam with an optimal size. This imaging +scheme is best suited for applications where lenses for the +biphoton wavelengths are less available and a high transverse +resolution is not required. We demonstrated that the proposed +scheme is analogous to the classical pinhole camera where the +biphoton source plays the role of the pinhole and derived its +spatial resolution and number of spatial modes. +See the supplementary material for further details of the an- +alytical model and examples with non-degenerate photons. + +5 +ACKNOWLEDGMENTS +We thank E. Santos and V. Gili for their insightful com- +ments. This work was supported by the Thuringian Ministry +for Economy, Science, and Digital Society; the European So- +cial Funds and the European Funds for Regional Develop- +ment (2017 FGR 0067, 2017 IZN 0012); the German Federal +Ministry of Education and Research (FKZ 13N14877, FKZ +03ZZ0434) and the Deutsche Forschungsgemeinschaft (DFG, +German Research Foundation, project ID 407070005). +DATA AVAILABILITY +The data that supports the findings of this study are avail- +able within the article and its supplementary material. +This article may be downloaded for personal use only. +Any other use requires prior permission of the author and +AIP Publishing. +This article appeared in A. Vega et al., +Appl. Phys. Lett. 117, 094003 (2020) and may be found +at https://doi.org/10.1063/5.0012477 +1D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, +and Y. H. Shih, +Phys. Rev. Lett. 74, 3600 (1995). +2T. B. Pittman, Y. H. Shih, D. V. Strekalov, +and A. V. Sergienko, +Phys. Rev. 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Lett. 123, 263602 (2019). + +arXiv:2301.00994v1 [quant-ph] 3 Jan 2023 +1 +(Dated: 4 January 2023) +1 + +I. +A. +1. +2 + diff --git a/59AzT4oBgHgl3EQfEfpg/content/tmp_files/load_file.txt b/59AzT4oBgHgl3EQfEfpg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0080535fd14f50ba1110255ea5dd5433865db29 --- /dev/null +++ b/59AzT4oBgHgl3EQfEfpg/content/tmp_files/load_file.txt @@ -0,0 +1,558 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf,len=557 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='00994v1 [quant-ph] 3 Jan 2023 Pinhole quantum ghost imaging Andres Vega,1, a) Sina Saravi,1 Thomas Pertsch,1, 2 and Frank Setzpfandt1 1)Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 15, 07745 Jena, Germany 2)Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 7, 07745 Jena, Germany (Dated: 4 January 2023) We propose a quantum ghost imaging scheme based on biphotons, that, by using a collimated pump beam of the right size for biphoton generation, obviates the need for lenses to achieve imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The scheme is found to be analogous to the classical pinhole camera, where we show that the equivalent to the classical pinhole size depends mainly on the width of the pump beam, but also on the thickness of the nonlinear crystal and the wavelengths of the biphoton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Quantum ghost diffraction1 and imaging2 rely on the spa- tial correlations of a biphoton, which can be created by the nonlinear process of spontaneous parametric down conversion (SPDC)3,4, where a pump (P) photon impinging on a crystal with second-order nonlinearity χ(2) is split into a pair of pho- tons called signal (S) and idler (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' After creation, the two photons are separated into two different paths and only one of them interacts with the object, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' the signal photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Then, the signal photon is measured by a detector with no spatial res- olution, whereas another detector with spatial resolution mea- sures the idler photon that never interacted with the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' None of the detectors alone can recover a diffraction pattern or image of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Remarkably, these can be retrieved by correlating the two measurements5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This measurement technique has two main advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' First, very low numbers of photons can be used due to the inherently better signal-to- noise ratio of quantum ghost imaging compared to imaging with classical light7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Additionally, ghost imaging with two- color biphotons can overcome limitations due to inaccessible wavelength ranges for illumination and detection9–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To form a ghost image, usually lenses are placed in the path of the signal and/or idler after the crystal2,7,8 or in the pump beam before the crystal12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The lenses introduce a parabolic phase-front in either of the beam paths, which results in the formation of the image in the coincidence measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' However, quantum ghost imaging can be also realized without lenses by adding the parabolic phase-front through engineer- ing the nonlinearity profile of the nonlinear crystal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' by using a nonlinear photonic crystal13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Furthermore, as ghost imaging can be also realized with classical light using ther- mal light sources14–19, their inherent property of acting like a phase-conjugated mirror16 in a ghost imaging scheme can also be used for lensless ghost imaging20,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In classical optics, lensless imaging can be also realized us- ing the principle of pinhole imaging22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In a pinhole camera, the object is located on one side of an opaque screen with a small pinhole, whereas the detector is on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' With- out the need of lenses, the detector captures a shadow of the object, which can be optimized by adapting the pinhole size22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Throughout this manuscript, we will refer to this shadow as an image although strictly no imaging is taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' For appli- a)Electronic mail: andres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='vega@uni-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='de cations where high spatial resolution is not needed, this type of lensless imaging has several advantages over imaging with lenses, among which are a larger depth of field, a wide angular field of view23, and its applicability in wavelength ranges for which high-quality lenses are less available24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In this work, we want to show that the advantages of pin- hole imaging can be also harnessed in quantum ghost imag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' For ghost imaging with thermal light, a pinhole-based scheme has been already proposed, where the optimal lensless imaging condition depends on the size of the thermal source26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We extend this approach of lensless imaging to the quantum regime with entangled photons based on the setup sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Here, we assume that the nonlinear crystal gener- ating photon pairs is illuminated by a collimated pump beam and, contrary to ghost imaging with thermal light, we use a bucket detector instead of a point detector behind the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We will show, that for specific pump beam diameters, pinhole quantum ghost imaging can be realized and we investigate its properties and optimum regime of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To this end, we start by discussing the biphoton joint spatial probability (JSP) and the quantum ghost pattern (G) together with a numeri- cal example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Later, we derive a simplified analytical model for our imaging scheme that explains the observations of the numerical example and allows the connection to the classical pinhole camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Using this model, we will furthermore dis- cuss the spatial resolution of pinhole quantum ghost imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Throughout this work, z is the propagation direction and we restrict our analysis, without loss of generality, to one transverse dimension x in position space, whose conjugate in momentum space is the transverse component of the wave- vector kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We also assume an infinitely extended nonlinear crystal in the transverse direction, which ensures transverse Nonlinear crystal Bucket detector Object Coincidence circuit Spatially resolving detector FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Sketch of the considered setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2 phase matching kxP = kxS + kxI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We assume an undepleted monochromatic classical pump beam of the form EP(x,z,t) = � dkxP φP(kxP) exp[i(kxPx+ kzPz− ωPt)], where φP(kxP) is the pump spatial spectrum, the longitudinal component of the wave-vector is kz = [(ωn/c)2 − k2 x]1/2 with ω/c = 2π/λ, λ being the wavelength in free space and ω the corresponding frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Additionally, we ignore the effects of the bound- aries between the crystal and its surrounding free space, which would cause refracted and reflected waves, and assume a sim- plified case with the crystal’s refractive index n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We in- vestigate signal and idler photons at fixed frequencies of ωI and ωS, such that ωP = ωS + ωI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This can be achieved exper- imentally by placing narrow bandpass filters centered around these frequencies in their beam paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Under these condi- tions, the biphoton quantum state after the filters will have the form27 |Ψ⟩ ∝ � dkxSdkxI ψSPDC(kxS,kxI) |kxS,ωS⟩|kxI,ωI⟩, with ψSPDC(kxS,kxI) = φP(kxS + kxI) sinc(∆kz lz/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Here, ∆kz = kzP − kzS − kzI, lz is the thickness of the crystal, and |kx,ω⟩ is the single-photon state defined by the transverse component of the wave-vector kx and frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The JSP of the biphoton state at the two detectors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 1 is28 JSP(xS,xI) ∝ ���F−1� hI(kxI)h2S(kxS) � to(kxS)∗ � h1S(kxS) ×ψSPDC(kxS,kxI) ������ 2 , (1) where F−1 is a two-dimensional inverse Fourier transform, (kxS,kxI) → (xS,xI), and h are free space transfer func- tions with hI(kxI) = exp(ikzI zI), h1S(kxS) = exp(ikzS d) and h2S(kxS) = exp[ikzS(zS − d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Finally, ∗ denotes the convo- lution only in kxS as the object with transmission To(xS) is in the signal arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The ghost pattern G(xI) for a bucket detector that collects all signal photons is derived from the JSP as G(xI) ∝ � dxS JSP(xS,xI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We assume a pump beam with Gaussian spatial spectrum, whose waist is located at the center of the nonlinear crystal at the plane z = 0, where φP ∝ exp[−σ2 P(kxS + kxI)2/2] has a flat wave front and a width σP in position space (see zoomed out region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' As shown in pinhole ghost imaging with a thermal source26, the size of the photon source has a similar role as the pin- hole size in classical optics, determining the optimal regime of imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This suggests that in the quantum regime, the same role can exist for the size of the biphoton source, which depends on the width of the pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To examine this premise, we numerically calculate quantum ghost imaging in a setup with a nonlinear crystal with thickness lz = 3 mm, a pump with wavelength of λP = 350 nm, degenerate down- converted photons with λS = λI = 700 nm, and detectors lo- cated at zS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='2 m and zI = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='5 m from the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' As ob- ject, we consider a double-slit with 940 µm slit separation, with unity transmission in each slit of 50 µm width and no transmission elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(a-d) we present the en- suing normalized JSPs calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (1) for different sizes of the pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The double-slit always results in two spots, whose separation is approximately five times larger than the slit separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Depending on the width of the pump beam, they change their widths and begin to overlap and in- terfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The interference is minimal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(c) with pump 0 1 (d) (e) (a) (c) 8 0 8 8 0 8 8 0 8 8 0 8 8 0 8 102 103 (b) (c) 0 1 (d) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (a-d) JSP(xS,xI) and corresponding (e) quantum ghost pat- tern G(xI) of a double-slit, 940 µm slit separation and 50 µm slit width, located at d = 30 cm produced by a pump width σP of (a) 58 µm, (b) 102 µm, (c) 167 µm, and (d) 800 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' width σP = 167 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(e), we show the corresponding ghost patterns, where the cases of (a-d) are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We see, that for a specific range of pump widths, two separate maxima are visible, corresponding to an image of the double-slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We note, that the well-known quantum ghost diffraction pattern1 of the object could be recovered using a large pump width, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(d), and replacing the bucket with a point detector which would measure only a horizontal cut through the JSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This numerical example portrays the core idea of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Other schemes12,13 have shown that a pump wave with a curved wave front, obtained by means of a lens placed be- fore the nonlinear crystal or using a photonic crystal, can also be used for quantum ghost imaging without lenses in the paths of the biphoton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(c) now shows, that lensless quantum ghost imaging can be also achieved by simply using a colli- mated pump beam with an optimal width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This is easily seen, as the Rayleigh length29 of the pump, 2πσ2 P/λP = 50 cm, is much larger than the crystal thickness, lz = 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To find the optimal conditions for this imaging scheme, we derive a simplified analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Here, we consider only one of the slits and assume it has infinitesimal width and is located at xS = a, which means that To(xS) = δ(xS − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This object is put into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (1) and in paraxial approxima- tion an analytical solution can be calculated (see supplemen- tary material), where we approximate the sinc function ap- pearing due to phasematching in the nonlinear crystal by a Gaussian30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We find a Gaussian ghost intensity pattern G(xI) ∝ exp[−(xI − x0)2/(2σ2 G)] with a width σG and a max- imum located at x0, given by σG = � 2Re � α−1 1 ��−1/2, (2) x0 = a Re � α−1 1 α2 �� Re � α−1 1 ��−1 , (3) where α1 = σ2 P + γ lz π (λI − λP)+ iλIzI 2π − � σ2 P − γ lzλP π � α2, (4) α2 = � σ2 P − γ lzλP π �� σ2 P + γ lz π (λS − λP)+ idλS 2π �−1 (5) 102 8 0 83 with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='455/4, a constant that comes from the sinc to Gaus- sian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Due to the bucket detector that collects all signal photons behind the object, the equations do not de- pend on zS, the distance of the object to the bucket detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' however, the model does depend on the location of the re- solving detector zI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' These distances remain at zS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='2 m and zI = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='5 m throughout the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3 shows the width σG of the ghost pattern and the normalized position x0/a with respect to the width of the pump σP and the distance of the ob- ject to the crystal d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3 that, for each object position d, the width of the ghost pattern σG has a minimum at a certain pump width σP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This confirms the observations of the numerical example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2 that uses d = 30 cm, where the optimal case for imaging, marked with a dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(c), leads to the narrowest ghost pattern for each of the slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In the rest of the cases, the pattern of each slit is too wide, result- ing in considerable overlap between them and hence the loss of visibility of the ghost pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' If a second infinitesimal slit would by at xS = −a, the distance between the two maxima in the ghost patterns would be 2x0, therefore, x0/a represents the magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3(b) verifies that for the numerical exam- ple in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2 the magnification is approximately equal to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' It also tells that the magnification is always negative, implying that the ghost image is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3(a) shows, that the minimum width of the ghost pat- tern becomes smaller as the object is placed farther from the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This value, however, does not reach zero, the behavior of σG converges to approximately the orange curve in the limit where the object is very distant from the crystal, d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Here, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (2) can be reduced to σ2 G = σ2 0 + σ−2 0 [zIλI/(4π)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The width of the ghost pattern of an infinitesimal slit object with the spatially resolving detector placed right after the crystal, σ0 = σG(zI = 0), is σ0 = �1 2σ2 P + γ � λI λS ��λPlz 2π ��1/2 , (6) an expression that depends only on the parameters of the biphoton source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' For a fixed value of zIλI, the ghost pattern width σG has a minimum value, namely σmin G = √ 2σ0, when σ2 0 = zIλI/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Remarkably, this result is equivalent to the optimal pinhole size σpinhole in a classical pinhole camera that creates the smallest point image of a slit upon spatially inco- herent illumination22, σ2 pinhole ∝ λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Hence, σ0 can be consid- ered the pinhole size of quantum ghost imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' It does not only depend on the width of the pump but also on the thickness of the crystal and the biphoton wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' However, the de- viation of the equivalent pinhole size σ0 from the pump width σP due to the biphoton wavelength is small as for a pump with negligible diffraction inside the crystal σ2 P ≫ λPlz/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Next, we analyze the magnification to complement the anal- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The magnification of the classical pinhole camera is given by geometric optics as −z/d with z the distance of the detector to the pinhole and d the distance of the object to the pinhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' A similar relation can be found from the analytical model of the proposed ghost imaging scheme using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (3), under the conditions that σ2 P ≫ (γ lz/π)(max(λS,λI) − λP) and {dλS/(2π), zIλI/(2π)} ≫ σ2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The first condition en- sures, that signal, idler and pump waves have negligible (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='4 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='8 0 200 400 45 30 15 0 0 200 400 d=3cm ↓ d=5cm d=10cm d=30cm d→∞ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (a) Width σG and (b) magnification of the position x0/a of the ghost pattern of a infinitesimal slit at xS = a with respect to the pump width σP at various distances d of the slit to the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The circle marks the parameters of the numerical example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' diffraction inside the nonlinear crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In this case, σP de- fines the size of the generated signal and idler beams inside the crystal, and hence their Rayleigh lengths are also much larger than the crystal thickness lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The second condition states that both the object and the resolving detector are in the far-field of the biphoton source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Under these conditions, following the steps detailed in the supplementary material, we find the mag- nification to be x0/a ≈ −(zI/d)(λI/λS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This is similar to the geometrical magnification of the classical pinhole camera but including again the biphoton wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' From this ex- pression, the magnification of the numerical example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2 can be quickly found −(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='5m/30cm)(700nm/700nm) = −5, see the supplementary material for an example with non- degenerate wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Noteworthy, our imaging scheme is not limited to the far-field domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We only take the approxi- mations to find the simple expression for the magnification to build the analogy with the classical pinhole camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We further harness the analytical model of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (2) and (3) to derive the transverse resolution of pinhole quantum imag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In the following, we use Rayleigh’s resolution criterion that is defined as the minimum distance between two point- like objects to distinguish them from one another29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' A smaller ghost pattern width σG of a point-like object results in a bet- ter distinction between neighboring objects, as was demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' At the same time, a larger magnification |x0/a| would results in a better distinction between two neigh- boring objects, as their images are further apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This suggests that, to optimally tell two objects apart, not only the ghost pat- tern width has to be taken into account but also the magnifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To test this hypothesis, we begin by taking the derived Gaussian ghost pattern from an infinitesimal slit A at xS = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' For the sake of simplicity we normalize its maximum to one, resulting in the ghost pattern GA = exp[−(xI − x0)2/(2σ2 G)], where σG is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (2) and x0 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' If we in- clude another similar slit B but at xS = −a, the resulting ghost pattern is symmetric with respect to xI = 0 and is given ap- proximately by G ≈ GA + GB, assuming negligible interfer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The two slits can be distinguished in this pattern when its visibility is above a certain threshold, here heuristically chosen to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' That means, the intensity at xI = 0 be- tween the two maxima should be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='4 of the maxi- mal intensity at xI = ±x0, which gives the threshold condition G(xI = 0)th ≈ GA(0)+GB(0) = 2GA(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Using this, the 4 0 100 200 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='7 1 d=3cm d=5cm d=10cm d=30cm � 0 1 0 1 10 0 10 (�� �� � ��� � d=1m 100 101 102 10-1 100 101 d ∞ d=3cm d=30cm d=1m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (a) Resolution R with respect to the pump width σP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The green line connects the minima of curves of a wider range of object distances d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (c) Number of spatial modes N with respect to the crystal thickness lz at various d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Numerically, JSP (top) and G (bottom) of a (b) double-slit and a (d) complex object (magnified transmission in yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Circles in (a) and (c) mark the parameters of (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' expression for GA, and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (2) and (3), we find the transverse spatial resolution R corresponding to the minimum resolvable distance 2a between two identical infinitesimal slits to be R = 2 � −ln[G(0)th/2]Re � α−1 1 ��1/2���Re � α−1 1 α2 ���� −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' (7) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(a) displays its dependence on the pump width σP at dif- ferent object distances d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The thick green line connects the minima of several curves over a wider range of d to portray the tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' One could naively expect from the width of the ghost pattern σG in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3(a) that the resolution R could improve as the object is farther from the crystal since the min- imum width becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' However, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(a) tells the opposite, as R is enhanced by the large magnification at small object distances d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This is confirmed numerically by consid- ering again the case depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(b), that uses a pump width of 102 µm and d = 30cm, and changing the position of the object to d = 10cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' The resulting JSP and ghost pattern G are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2(b), the visibility is increased due to the larger magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This originates from the fact that, for a fixed object distance d, the minimum ghost pattern width σG does not coincide with the largest magnifi- cation x0/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Hence, the optimal pump width σP that results in the best resolution is not necessarily when σG is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This is only true for larger values of d where x0/a is almost independent of σP, as in the numerical example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Noteworthy, the green tendency line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(a) hints that the closer the object is to the crystal and the smaller the pump width, the better the resolution R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' However, the smaller the pump width, the broader its spatial spectrum becomes, in such case the used paraxial approximation does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' A non- paraxial formulation is a matter of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Addition- ally, R will depend linearly on the object position d for large values of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This is because in such case, the ghost pattern width σG is nearly independent of d, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 3(a), and the magnification is |x0/a| ∝ 1/d, as already explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' In addition to the resolution, it is also of great interest to describe the extent of the signal illumination onto the object, which limits the object size that can be imaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This is finite and can be found from a projection of the JSP at the object po- sition onto the signal axis (see the analytical expression in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This illumination size σS increases with the position of the object d and decreases with the crystal thickness lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Importantly, the ratio between the illumination size and the resolution tells the maximum number of identical infinitesimal slits that can be resolved inside the illuminated region of the object, N ≡ σS/R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' describes the number of independent spatial modes in the object illumination31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Its de- pendence on the crystal thickness lz is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(c) for various d, each at a pump width σP that minimizes the res- olution as described by the green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To increase the number of spatial modes N, a thinner crystal can be used, similar to other quantum imaging schemes2,32, as it allows a larger range of transverse wave-vectors33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Also, the object could be put farther away from the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Here, the increase of N with d has a limit due to the linear dependence of both σS and R on d for very distant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Finally, we found that photon-pairs with non-degenerate wavelengths show a minor improvement in the resolution and number of modes, see the supplementary material for some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Lastly, we sum up the main features of the proposed setup with a ghost image of a more complicated object with vari- ous shapes and transmissions, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This object is placed at d = 1 m and we use a pump width σP = 258 µm that optimizes the ghost image resolution to R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='5 mm, allows N = 10 spatial modes and has a magnification x0/a = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Moreover, unlike the setup with a pseudo-thermal source26, we see in the JSP of this example that an integrating bucket detector is necessary to show the whole illuminated section of the object in the ghost pattern, a point detector would not be able to detect signals from all objects as it would just “take a horizontal thin slice” of the JSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' To conclude, we proposed a quantum ghost imaging scheme without lenses in the biphoton arms by means of a collimated pump beam with an optimal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This imaging scheme is best suited for applications where lenses for the biphoton wavelengths are less available and a high transverse resolution is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' We demonstrated that the proposed scheme is analogous to the classical pinhole camera where the biphoton source plays the role of the pinhole and derived its spatial resolution and number of spatial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' See the supplementary material for further details of the an- alytical model and examples with non-degenerate photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 5 ACKNOWLEDGMENTS We thank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Santos and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Gili for their insightful com- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This work was supported by the Thuringian Ministry for Economy, Science, and Digital Society;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' the European So- cial Funds and the European Funds for Regional Develop- ment (2017 FGR 0067, 2017 IZN 0012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' the German Federal Ministry of Education and Research (FKZ 13N14877, FKZ 03ZZ0434) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, project ID 407070005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' DATA AVAILABILITY The data that supports the findings of this study are avail- able within the article and its supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' This article may be downloaded for personal use only.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' A 71, 013801 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 17R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Bennink, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' Bentley, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' W.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} +page_content=' 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf'} diff --git a/5NFAT4oBgHgl3EQfmh0R/content/tmp_files/2301.08623v1.pdf.txt b/5NFAT4oBgHgl3EQfmh0R/content/tmp_files/2301.08623v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cc09c088cf5cf90c634930367054ce831c960b0 --- /dev/null +++ b/5NFAT4oBgHgl3EQfmh0R/content/tmp_files/2301.08623v1.pdf.txt @@ -0,0 +1,3346 @@ +ERGODIC PROPERTIES OF A PARAMETERISED FAMILY OF SYMMETRIC +GOLDEN MAPS: THE MATCHING PHENOMENON REVISITED +KARMA DAJANI AND SLADE SANDERSON +Abstract. We study a one-parameter family of interval maps {Tα}α∈[1,β], with β the golden mean, defined +on [−1, 1] by Tα(x) = β1+|t|x − tβα where t ∈ {−1, 0, 1}. For each Tα, α > 1, we construct its unique, +absolutely continuous invariant measure and show that on an open, dense subset of parameters α, the +corresponding density is a step function with finitely many jumps. We give an explicit description of the +maximal intervals of parameters on which the density has at most the same number of jumps. A main tool +in our analysis is the phenomenon of matching, where the orbits of the left and right limits of discontinuity +points meet after a finite number of steps. Each Tα generates signed expansions of numbers in base 1/β; via +Birkhoff’s ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies +of digits in generic Tα-expansions. +In particular, the frequency of 0 is shown to vary continuously as a +function of α and to attain its maximum 3/4 on the maximal interval [1/2 + 1/β, 1 + 1/β2]. +1. Introduction +Dynamical systems given by piecewise monotone maps T : I → I of an interval have a rich history: +besides having applications in various fields—including population ecology ([3]) and controlled switching +circuits ([1])—these systems are often used to produce expansions of numbers from the underlying interval +I. +Examples include decimal, n-ary, continued fraction, (generalised) L¨uroth and β-expansions, though +this list is far from exhaustive. +A common theme in the study of these expansions is the investigation +of asymptotic relative frequencies of digits occurring in typical (i.e. Lebesgue–almost all) expansions. To +this end, the standard procedure is the construction of an ergodic, T-invariant measure µ equivalent to +Lebesgue measure λ and a calculation of the µ-measure of the subinterval of I corresponding to the digit(s) +in question. +Birkhoff’s ergodic theorem asserts that the measure of this subinterval equals the desired +asymptotic frequency. +In [13], invariant measures and frequencies of digits are studied for a family of symmetric doubling maps +{Dη}η∈[1,2] defined on [−1, 1] by Dη(x) = 2x − d(x)η with d(x) ∈ {−1, 0, 1}. These maps produce signed +binary expansions of numbers x ∈ [−1, 1] of the form x = η � +n≥1 dn/2n with each dn ∈ {−1, 0, 1}. It is shown +that each Dη, η > 1, admits an ergodic, invariant measure equivalent to Lebesgue measure. The authors use +a curious property called matching—defined in the sequel—to prove that there is a countable collection of +disjoint, open subintervals of [1, 2] whose union has full measure, and such that on each such subinterval, the +densities of the corresponding invariant measures are step functions with at most the same, finite number of +jumps. These explicitly constructed measures are then used to study the asymptotic frequency of the digit +0 in generic expansions. This frequency is shown to be continuous as a function of η and attains a maximal +value of 2/3 on the maximal interval [6/5, 3/2]. Moreover, the frequency function is either constant, strictly +increasing or strictly decreasing on each of the aforementioned subintervals of [1, 2]. +The present article continues these themes of inquiry with a parameterised family of skewed symmetric +golden maps {Tα}α∈[1,β], with β = ( +√ +5 + 1)/2 the golden mean, i.e. the positive real solution to β2 = β + 1. +Department of Mathematics, Utrecht University, P.O. Box 80010, 3508TA Utrecht, The Netherlands +E-mail addresses: k.dajani1@uu.nl and s.b.sanderson@uu.nl. +Date: January 20, 2023. +2020 Mathematics Subject Classification. 37E05 (Primary) 28D05, 37A05 (Secondary). +Key words and phrases. invariant measure, ergodic theory, matching, interval map, number expansions, digit frequency. +1 +arXiv:2301.08623v1 [math.DS] 20 Jan 2023 + +Each Tα : [−1, 1] → [−1, 1] is defined by +Tα(x) := +� +� +� +� +� +β2x + βα, +x ∈ [−1, −1/β) +βx, +x ∈ [−1/β, 1/β] +β2x − βα, +x ∈ (1/β, 1] +; +see Figure 1. Setting J−1 := [−1, −1/β), J0 := [−1/β, 1/β] and J1 := (1/β, 1], the map Tα may be written +more succinctly as +Tα(x) = β1+|t(x)|x − t(x)βα, +(1) +where t(x) ∈ {−1, 0, 1} is the unique index for which x ∈ Jt(x). For j ≥ 1, set tα,j(x) := t(T j−1 +α +(x)); the +sequence of digits (tα,j(x))j≥1 ∈ {−1, 0, 1}N records indices of the subsequent subintervals J−1, J0 or J1 +entered by the forward orbit of x. With this notation, equation (1) gives for each j ≥ 1 +T j +α(x) = β1+|tα,j(x)|T j−1 +α +(x) − tα,j(x)βα. +Solving this for T j−1 +α +(x), induction shows that for any n ≥ 1, +x = α +n +� +j=1 +tα,j(x) +βj−1+�j +k=1 |tα,k(x)| + +T n +α (x) +βn+�n +k=1 |tα,k(x)| . +Taking the limit n → ∞ and recalling that |T n +α (x)| ≤ 1 gives +x = α +� +j≥1 +tα,j(x) +βj−1+�j +k=1 |tα,k(x)| . +Note that for fixed α, this process determines a unique expansion for each x ∈ [−1, 1]. We refer to both this +expansion and the corresponding sequence of digits (tα,j(x))j≥1 as the Tα-expansion of x. +Phenomena analogous to those observed in [13] are found to occur for the skewed symmetric binary maps +Tα. In particular, we prove: +Theorem 1.1. For each α ∈ (1, β], the map Tα has a unique—hence ergodic—absolutely continuous in- +variant probability measure µα. Moreover, µα is equivalent to Lebesgue measure λ, and there is a countable +collection {Id}d∈M of disjoint open subintervals of [1, β] of full Lebesgue measure, such that for fixed d ∈ M +the density of each µα with α ∈ Id is a step function with at most the same, finite number of jumps. +Via Birkhoff’s ergodic theorem, these measures are employed to show the following: +Theorem 1.2. The asymptotic relative frequency of the digit 0 in Lebesgue-a.e. +Tα-expansion depends +continuously on α ∈ [1, β] and attains a maximum value of 3/4 on the (maximal) interval [1/2+1/β, 1+1/β2]. +Furthermore, the frequency function is either constant, strictly increasing or strictly decreasing on each Id. +As in [13], the main tool used to construct the Tα-invariant measures is a property called matching. An +interval map T : I → I is said to have matching if for each critical point c ∈ I, the orbits of the left and +right limits y± := limx→c± T(x) agree after some finite number of steps.1 That is, for each critical point +c ∈ I there are integers M, N ≥ 0 for which T M(y−) = T N(y+). +Matching has gained considerable attention in recent years. Intricacies of the metric entropy function +of Nakada’s α-continued fraction maps have been studied using matching in [20], [7], [8], [18], [2] and [9]. +In particular, matching is used in [18] to determine the natural extension for each α-continued fraction +transformation, and it is shown that the set of α ∈ [0, 1] for which matching does not occur has zero +Lebesgue measure. The Lebesgue measure of this set of non-matching parameters—in addition to the fact +that its Hausdorff dimension is 1—is also shown in [8]. Matching is used in [16] to determine invariant +measures for the related family of α-Rosen continued fraction transformations. A parameterised family of +linear maps with one increasing and one decreasing branch are considered in [4], and matching is used to show +that in some parameter regions, the Lyapunov exponent and topological entropy are constant. A geometric +explanation of matching for a similar family of maps is given in [12], and further implications of matching for +these maps—including smoothness of entropy on an open dense subset of parameters—is considered in [6]. +1Some authors require that the one-sided derivatives also agree at these times, in which case the map may be said to have +strong matching ([15]). This extra condition is not needed for our purposes. +2 + +The notion of matching is extended to random dynamical systems in [15] and is used to study the asymptotic +frequency of the digit 0 in typical signed binary expansions arising from a family of random interval maps. +Matching has also been investigated for generalised β-transformations, a certain class of continued fraction +expansions with finite digit sets, and Lorenz maps (see [5], [10] and [11], respectively). +The present paper exploits the phenomenon of matching in a fashion similar to that of [13]. There the +authors use results of [17], which gives formulas for densities of the absolutely continuous invariant measures +of piecewise linear expanding interval maps. These densities are—in general—infinite sums of (finite) step +functions which are determined by the orbits of the left and right limits at critical points of the underlying +interval map. However, when matching occurs the infinite sum becomes finite, and the density itself is a +finite step function depending only on these orbits before matching. +In [13], it is shown that matching +occurs for the symmetric doubling map Dη on a set of parameters η in [1, 2] of full Lebesgue measure. For +these matching parameters, the orbits of the left and right limits at the critical points before matching are +studied in detail, and this information is used to provide an explicit formula for the density of the (unique) +absolutely continuous invariant probability measure for each Dη with matching. The parameter space [1, 2] +is divided into a countable union of (maximal) open intervals—called matching intervals—where each Dη +has matching, and a Lebesgue-null set of non-matching parameters with Hausdorff dimension 1. On each +matching interval, matching occurs after the same number of steps, and for each left/right limit at a critical +point, the digits of the corresponding signed binary expansions agree before matching. +While the results of the present paper imply that the same direct approach of understanding matching +for the skewed symmetric golden maps Tα can be applied to construct the invariant measures asserted in +Theorem 1.1, we find that the unequal slopes of the different branches present difficulties. To circumvent +these, we instead study matching for a family of symmetric golden maps {Sα}α∈[1,β] of constant slope for +which the skewed symmetric golden maps {Tα}α∈[1,β] are jump transformations, and it is subsequently shown +that the parameters α for which the maps Tα and Sα have matching coincide (Proposition 2.9). Equipped +with this result, one could then use the formulas from [17] to determine invariant densities for the Tα with +matching; however, we proceed in the simpler setting of the symmetric maps Sα—determining invariant +densities and the frequencies of digits for these—and finally use the fact that Tα is the jump transformation +of Sα to determine invariant measures and frequencies of digits for the original skewed symmetric golden +maps. +The paper is organised as follows. In §2 the symmetric golden maps {Sα}α∈[1,β] are introduced. These +are shown in §2.1 to have matching for Lebesgue–a.e. α ∈ [1, β], and we also prove here that the matching +parameters of both families {Sα}α∈[1,β] and {Tα}α∈[1,β] coincide. Subsections 2.2 and 2.3 are devoted to +understanding the finer structure of the set of matching parameters. The former provides a classification of +all matching intervals and of the orbits of all left and right limits at critical points before matching occurs. +In the latter, it is shown that all (but two) of the matching intervals generate in a natural fashion a whole +‘cascade’ of countably many matching intervals with adjacent endpoints. In §3 we use the results of the +preceding section to prove Theorems 1.1 and 1.2. In particular, explicit formulas for densities of the unique, +absolutely continuous invariant measures of the symmetric golden maps Sα are provided in §3.1, and the +invariant measures of the skewed maps Tα are expressed in terms of these. These measures are used in §3.2 +to determine expressions for the asymptotic frequencies of the digit 0 in typical Sα- and Tα-expansions. The +maximal frequencies of the digit 0 as functions of α are considered in §3.3.. Proofs of some technical results +are provided in an appendix (§4). +Acknowledgments. This work is part of project number 613.009.135 of the research programme Mathe- +matics Clusters which is financed by the Dutch Research Council (NWO). +2. Symmetric golden maps Sα +As mentioned in §1, we determine invariant measures and the frequencies of digits for a family of symmetric +golden maps {Sα}α∈[1,β] for which the {Tα}α∈[1,β] are jump transformations. These invariant measures and +frequencies are then used to determine the invariant measures and frequencies of digits for the original Tα. +The maps Sα are defined as follows: for α ∈ [1, β], let Sα : [−1, 1] → [−1, 1] be given by +Sα(x) := βx − t(x)α, +3 + +-1 +−1/β +0 +1/β +1 +-1 +0 +1 +Figure 1. The maps Tα (blue) and Sα (red) with α = 1.4. Note that Tα = Sα on the +middle interval J0 = [−1/β, 1/β]. +with t(x) ∈ {−1, 0, 1} as in §1; see Figure 1. Note that Sα(x) ∈ J0 for each x ∈ J−1 ∪ J1. Using this, one +readily verifies that +Tα(x) = +� +Sα(x), +x ∈ J0 +S2 +α(x), +x ∈ J−1 ∪ J1 +, +(2) +i.e. Tα is the jump transformation for Sα with respect to the sweep-out set J0 = [−1/β, 1/β] (see, e.g. §11.4 +of [14]). For each j ≥ 1, let sα,j(x) := t(Sj−1 +α +(x)). With induction one finds that for each k ≥ 0, +Sk +α(x) = βk +� +�x − α +k +� +j=1 +sα,j(x)/βj +� +� +(3) +(with the summation for k = 0 understood to be 0). Since |Sk +α| ≤ 1, dividing both sides by βk and taking +the limit as k approaches infinity gives +x = α +� +j≥1 +sα,j(x)/βj. +(4) +Following our convention from §1, we refer to both the right-hand side of Equation (4) and the corresponding +sequence (sα,j(x))j≥1 of digits in {0, ±1}N as the Sα-expansion of x. Again this process determines—for +fixed α—a unique expansion for each x ∈ [−1, 1]; moreover, if x, y ∈ [−1, 1] have the same Sα-expansion, +then Equation (3) can be used to show that x = y. Also note that not every sequence in {0, ±1}N is an +Sα-expansion; in particular, a 1 or −1 is necessarily followed by a 0. +As the orbits of 1 and 1−α will be studied in detail below, we fix special notation for their Sα-expansions: +let dα,j := sα,j(1) and eα,j := sα,j(1 − α) for each α ∈ [1, β] and j ≥ 1. When α is understood, it is +suppressed from the notation, and we simply write dj := dα,j and ej := eα,j. +2.1. Matching almost everywhere. In this section, we show that the maps Sα (and Tα) have matching +on a set of full Lebesgue measure.2 The map Sα has two critical points ±1/β. Due to symmetry, it suffices +to consider the matching criteria only for the positive critical point 1/β. Note that limx→1/β− Sα(x) = 1 +and limx→1/β+ Sα(x) = 1 − α. Hence Sα has matching if and only if there are integers M, N ≥ 1 for which +SM +α (1) = SN +α (1 − α). +We begin by investigating matching in a number of specific cases. First, note that 1 ∈ J1 and 1 − α ∈ J0 +for all α ∈ [1, β]. +2The general approach to proving this result largely follows that of §2.2 of [13]; however, we shall see that the dynamics of +the symmetric golden maps Sα are—in a sense—more delicate than those of the previously studied symmetric binary maps +(compare, e.g. Proposition 2.1 below with Proposition 2.1 of [13]). +4 + +(i) If α ∈ (1 + 1/β2, β], then +Sα(1) = β − α ∈ [0, 1/β3) ⊂ J0, +Sα(1 − α) = β − βα ∈ [−1, −1/β) ⊂ J−1, +S2 +α(1) = β2 − βα ∈ J0 +and +S2 +α(1 − α) = β2 − β2α + α = β2 − βα ∈ J0 +shows that Sα has matching with M = N = 2. +(ii) If α = 1 + 1/β2, then +Sα(1) = β − α = 1/β3 ∈ J0, +Sα(1 − α) = β − βα = −1/β ∈ J0, +S2 +α(1) = 1/β2 ∈ J0, +S2 +α(1 − α) = −1 ∈ J−1, +S3 +α(1) = 1/β ∈ J0, +S3 +α(1 − α) = −1/β3 ∈ J0, +S4 +α(1) = 1 ∈ J1 +and +S4 +α(1 − α) = −1/β2 = 1 − α ∈ J0, +so Sα has a Markov partition, namely +� +[−1/β3/1/β3], ±(1/β3, 1/β2], ±(1/β2, 1/β], ±(1/β, 1] +� +, +and no matching. +(iii) If α ∈ (1 + 1/β3, 1 + 1/β2), +Sα(1) = β − α ∈ (1/β3, 1/β2) ⊂ J0, +Sα(1 − α) = β − βα ∈ (−1/β, −1/β2) ⊂ J0, +S2 +α(1) = β2 − βα ∈ (1/β2, 1/β) ⊂ J0, +S2 +α(1 − α) = β2 − β2α ∈ (−1, −1/β) ⊂ J−1, +S3 +α(1) = β3 − β2α ∈ (1/β, 1) ⊂ J1, +S3 +α(1 − α) = β3 − (β3 − 1)α ∈ (−1/β3, 1/β3) ⊂ J0, +S4 +α(1) = β4 − (β3 + 1)α ∈ J0 +and +S4 +α(1 − α) = β4 − (β4 − β)α ∈ J0. +Since β4 − β3 = β2 = β + 1, we find that S4(1) = S4(1 − α), so Sα has matching with M = N = 4. +(iv) If α = 1 + 1/β3, +Sα(1) = β − α = 1/β2 ∈ J0, +Sα(1 − α) = β − βα = −1/β2 ∈ J0, +S2 +α(1) = 1/β ∈ J0, +S2 +α(1 − α) = −1/β ∈ J0, +S3 +α(1) = 1 ∈ J1, +S3 +α(1 − α) = −1 ∈ J−1 +and +S4 +α(1 − α) = −1/β2 ∈ J0, +so Sα has a Markov partition and no matching. +(v) If α ∈ (1, 1 + 1/β3), then +Sα(1) = β − α ∈ (1/β2, 1/β) ⊂ J0, +Sα(1 − α) = β − βα ∈ (−1/β2, 0) ⊂ J0, +S2 +α(1) = β2 − βα ∈ (1/β, 1) ⊂ J1, +S2 +α(1 − α) = β2 − β2α ∈ (−1/β, 0) ⊂ J0, +S3 +α(1) = β3 − (β2 + 1)α ∈ (−1/β3, 1/β) ⊂ J0 +and +S3 +α(1 − α) = β3 − β3α ∈ (−1, 0) ⊂ J−1 ∪ J0. +This case will be considered more closely in what follows. +(vi) If α = 1, then Sα(1) = 1/β ∈ J0, S2 +α(1) = 1 ∈ J1 and Sα(1 − α) = 0 = 1 − α ∈ J0. Thus there is a +Markov partition and no matching. +Note that in the cases above in which there is matching—namely (i) and (iii)—we have M = N (a property +called neutral matching in [6]). We shall see below that this is always the case, i.e. Sα has matching if and +only if there is some m ≥ 1 for which Sm +α (1) = Sm +α (1−α). For this we need the following proposition—key to +a number of arguments throughout—which states that the difference between subsequent points in the orbits +of 1 and 1 − α can take on at most four values. Recall that (dj)j≥1 and (ej)j≥1 denote the Sα-expansions of +1 and 1 − α, respectively. +Proposition 2.1. For every α ∈ [1, β] and j ≥ 0, +Sj +α(1) − Sj +α(1 − α) ∈ {0, α/β, α, βα}. +Proof. For α /∈ (1, 1 + 1/β3), the statement is verified with the cases above, so assume α ∈ (1, 1 + 1/β3). We +use induction on j. The result clearly holds for j = 0; assume for some j = k − 1 ≥ 0 that +Sk−1 +α +(1) − Sk−1 +α +(1 − α) = y +5 + +for some y ∈ {0, α/β, α, βα}. If y = 0, then also Sj +α(1) − Sj +α(1 − α) = 0 for all j ≥ k − 1. Suppose y ̸= 0, +and note that +Sk +α(1) − Sk +α(1 − α) = (βSk−1 +α +(1) − dkα) − (βSk−1 +α +(1 − α) − ekα) = βy − (dk − ek)α. +We determine the difference above for each y ∈ {α/β, α, βα}: +(i) y = α/β: Since 1/β < y < 2/β, we have (dk, ek) = (1, 0), (0, −1) or (0, 0). In the first two cases +Sk +α(1) − Sk +α(1 − α) = 0, +and in the third +Sk +α(1) − Sk +α(1 − α) = α. +(ii) y = α: Since 1/β < y < 1 + 1/β3 = 2/β, we again have (dk, ek) = (1, 0), (0, −1) or (0, 0). In the first +two cases +Sk +α(1) − Sk +α(1 − α) = βα − α = α/β, +and in the third +Sk +α(1) − Sk +α(1 − α) = βα. +(iii) y = βα: Since y > 2/β, we must have (dk, ek) = (1, −1), and hence +Sk +α(1) − Sk +α(1 − α) = β2α − 2α = α/β. +□ +The previous proposition can be used to give an equivalent definition of matching: +Proposition 2.2. The map Sα has matching if and only if there is some m ≥ 1 for which Sm +α (1) = Sm +α (1−α). +Proof. One direction is immediate; for the other, suppose there are distinct M, N ≥ 1 for which SM +α (1) = +SN +α (1 − α). Assume for the sake of contradiction that Sj +α(1) ̸= Sj +α(1 − α) for all j ≥ 1. By Proposition 2.1, +Sj +α(1) − Sj +α(1 − α) ≥ α/β ≥ 1/β, +and hence +Sj +α(1 − α) ≤ Sj(1) − 1/β ≤ 1 − 1/β = 1/β2 +for each j. If Sj +α(1 − α) ∈ (0, 1/β2], then there is some k ≥ 0 for which Sj+k +α +(1 − α) = βkSj +α(1 − α) > 1/β2, +contradicting the above, and thus Sj +α(1−α) ≤ 0 for each j. A similar argument implies Sj +α(1) ≥ 0 for each j. +But SM +α (1) = SN +α (1 − α), so this common value must be 0. Since 0 is fixed by Sα, we have the contradiction +that Sm +α (1) = 0 = Sm +α (1 − α) with m = max{M, N}. +□ +We can now define a canonical index to describe when matching occurs: +Definition 2.1. The matching index of Sα is +m(α) := inf{m ≥ 1 | Sm +α (1) = Sm +α (1 − α)} ∈ N ∪ {∞}. +The cases above together with the proof of Proposition 2.1 reveal a strong interdependence between the +orbits of 1 and 1 − α, which is summarised in the graph of Figure 2. In particular, note that if matching +occurs with matching index m := m(α), then Sm−1 +α +(1)−Sm−1 +α +(1−α) = α/β and (dm, em) ∈ {(1, 0), (0, −1)}. +Since Sα-expansions cannot contain consecutive non-zero digits, this implies Sm−2 +α +(1) − Sm−2 +α +(1 − α) = α +and (dm−1, em−1) ∈ {(1, 0), (0, −1)}. +For m > 2, this further implies Sm−3 +α +(1) − Sm−3 +α +(1 − α) = α/β +and (dm−2, em−2) = (0, 0). Thus if Sα has matching with index m > 2, then the final three digits of the +Sα-expansions of 1 and 1 − α before matching are given by +�dm−2dm−1dm +em−2em−1em +� +∈ +��010 +001 +� +, +�001 +010 +�� +, +(5) +where w := −w for w ∈ {0, ±1}. Conversely, if for some m > 2, three consecutive digits of the Sα-expansions +of 1 and 1 − α are given by (5), then the proof implies that Sα has matching with index m. +A number of characterisations of matching for Sα can be derived from Proposition 2.1 and Figure 2. For +these we fix some notation: for each x ∈ [−1, 1] and α ̸= 1, let +ℓα(x) := inf +j≥0{Sj +α(|x|) ≤ 0} − 1, +6 + +0 +α/β +α +βα +� 0 +1 +� +� 0 +0 +� +� 1 +0 +� +� 0 +0 +� +� 0 +0 +� +� 1 +0 +� +� 0 +0 +� +� 0 +1 +� +� 0 +0 +� +� 0 +0 +� +� 1 +0 +� +� 0 +0 +� +� 1 +0 +� +� 0 +1 +� +� 0 +1 +� +� 1 +0 +� +� 0 +1 +� +� 0 +0 +� +� 0 +0 +� +� 1 +1 +� +� 1 +1 +� +� 0 +0 +� +� dj +dj +� +� dj+1 +dj+1 +� +Figure 2. A graphical representation of the interdependence of the orbits of 1 and 1 − α +for α ∈ [1, β]. Vertices represent the differences Sj−1 +α +(1) − Sj−1 +α +(1 − α) for j ≥ 1, and the +beginnings and ends of edges are marked +� dj +ej +� +and +� dj+1 +ej+1 +� +, respectively, where w := −w for +w ∈ {0, ±1}. Cyan edges are taken if and only if Sα has matching. +and set +ℓα := min{ℓα(1), ℓα(1 − α)}. +Lemma 2.3. For α ̸= 1, Sα has matching if and only if ℓα < ∞. Moreover, if ℓα < ∞, then m(α) ∈ +{ℓα + 1, ℓα + 2}. +Proof. Let ℓ := ℓα. That matching implies ℓ < ∞ is immediate. Now suppose ℓ < ∞, and assume without +loss of generality that ℓ = ℓα(1 − α) and thus Sℓ+1 +α +(1 − α) ≥ 0 (the other case is similar). The definitions of +ℓ and m(α) give ℓ + 1 ≤ m(α). By Proposition 2.1, Sℓ+1 +α +(1 − α) ≥ 0 and α > 1 imply +Sℓ+1 +α +(1) − Sℓ+1 +α +(1 − α) ∈ {0, α/β}. +The result holds if the difference is 0. If the difference is α/β, we must have (dℓ+2, eℓ+2) = (1, 0). From +Figure 2, this implies +Sℓ+2 +α +(1) − Sℓ+2 +α +(1 − α) = 0. +□ +Corollary 2.4. For α ̸= 1, Sα has matching if and only if there exists some j ≥ 1 such that +Sj +α(1) ∈ (1/β, α/β] +or +Sj +α(1 − α) ∈ [−α/β, −1/β). +Moreover, ℓα(1) and ℓα(1 − α), respectively, are the infimums over all j for which the above inclusions hold. +Proof. This follows from Lemma 2.3 and the facts that +S−1 +α ([−1, 0]) ∩ (0, 1] = (1/β, α/β] +and +S−1 +α ([0, 1]) ∩ [−1, 0) = [−α/β, 1/β). +□ +Due to symmetry, the above corollary states that Sα has matching if and only if the orbit of either 1 or of +α−1 enters the region (1/β, α/β]. We shall see that this occurs for Lebesgue–a.e. α by relating the beginnings +of these orbits to the beginnings of certain orbits of the (ergodic) β-transformation B : [0, 1] → [0, 1] defined +by B(x) = βx (mod 1). Set +b(x) := +� +0, +x < 1/β +1, +x ≥ 1/β , +7 + +and for each j ≥ 1, let +bj(x) := b(Bj−1(x)). +We call the sequence (bj(x))j≥1 the β-expansion (also referred to as the greedy-expansion) of x. Via induction, +one finds that for each k ≥ 0, +Bk +α(x) = βk +� +�x − +k +� +j=1 +bj(x)/βj +� +� . +(6) +Lemma 2.5. Let x ∈ {1, α − 1}, α ̸= 1. Then +(i) Sj +α(x) = αBj(x/α) for each 0 ≤ j ≤ ℓα(x), +(ii) sα,j(x) = bj(x/α) for each 1 ≤ j ≤ ℓα(x) and +(iii) ℓα(x) is the infimum over all j for which Bj(x/α) ∈ (1/βα, 1/β]. +Proof. Claim (iii) will follow from claim (i), Corollary 2.4 and the fact that ℓα(x) = ℓα(−x). We prove claim +(i) via induction on j. Certainly Sj +α(x) = αBj(x/α) for j = 0. Now suppose this equality holds for some +j = k − 1 with 0 ≤ k − 1 < ℓα(x). By Corollary 2.4, Sk−1 +α +(x) ∈ [0, 1]\(1/β, α/β], and we find +Sk +α(x) = +� +βSk−1 +α +(x), +Sk−1 +α +(x) ∈ [0, 1/β] +βSk−1 +α +(x) − α, +Sk−1 +α +(x) ∈ (α/β, 1] += +� +βαBk−1(x/α), +Bk−1 +α +(x/α) ∈ [0, 1/βα] +βαBk−1(x/α) − α, +Bk−1 +α +(x/α) ∈ (1/β, 1/α] += αBk(x/α), +so the first claim holds. Furthermore, the equality in (i) gives for each 1 ≤ j ≤ ℓα(x) that Sj−1 +α +(x) ∈ [0, 1/β] +if and only if Bj−1(x/α) ∈ [1, 1/βα] and Sj−1 +α +(x) ∈ (α/β, 1] if and only if Bj−1(x/α) ∈ (1/β, 1/α]. Thus +sα,j(x) = bj(x/α) for such j, proving claim (ii). +□ +Corollary 2.4, Lemma 2.5 and symmetry of Sα give yet another characterisation of matching in terms of +the map B: +Corollary 2.6. For α ̸= 1, Sα has matching if and only if there exists some j ≥ 0 such that +Bj(1/α) ∈ (1/βα, 1/β] +or +Bj(1 − 1/α) ∈ (1/βα, 1/β]. +Moreover, ℓα(1) and ℓα(1 − α), respectively, are the infimums over all j for which the above inclusions hold. +The previous results together with ergodicity of B can now be used to prove that Sα has matching for a +set of parameters α of full Lebesgue measure. The proof is nearly identical to that of Proposition 2.3 of [13] +but is included here for the ease of the reader. +Proposition 2.7. The map Sα has matching for Lebesgue–a.e. α ∈ [1, β]. +Proof. Let α ∈ (1, β] and k ∈ N with k > β3. By ergodicity of B with respect to Lebesgue measure (§4 of +[22]), for Lebesgue–a.e. x ∈ [0, 1] there exists some j ≥ 1 such that Bj(x) ∈ (1/β − 1/k, 1/β]. Note that +1/βα < 1/β − 1/k if and only if α > k/(k − β). Thus for Lebesgue–a.e. α ∈ (k/(k − β), β], there exists some +j ≥ 1 such that +Bj(1/α) ∈ (1/β − 1/k, 1/β] ⊂ (1/βα, 1/β]. +By Corollary 2.6, Sα has matching for Lebesgue–a.e. α ∈ (k/(k − β), β]. Let Ak denote the set of α ∈ +(k/(k − β), β] for which Sα does not have matching. Then ∪k>β3Ak has Lebesgue measure 0 and equals the +set of all α ∈ (1, β] for which Sα does not have matching. +□ +The finer structure of the set of matching parameters α ∈ [1, β] is considered in §§2.2 and 2.3 below. +Before investigating this structure, we show that matching occurs for Sα if and only if it occurs for the +corresponding jump transformation Tα. The following lemma may be deduced from the general theory of +jump transformations, but a proof is included for completeness. +8 + +Lemma 2.8. Fix x ∈ [−1, 1] and let j1 < j2 < j3 < . . . be an enumeration of the set +{j ≥ 0 | Sj +α(x) ∈ J0}. +Then T k +α(x) = Sjk+1 +α +(x) for all k ≥ 1. +Proof. The claim is immediate for k = 1 by (2) and the fact that Sα(J−1 ∪ J1) ⊂ J0. Now suppose the +result holds for some k ≥ 1, and let i ∈ {0, 1} be minimal such that Si +α(Sjk+1 +α +(x)) ∈ J0. By definition, then, +jk+1 = jk + i + 1, and +T k+1 +α +x = Tα(Sjk+1 +α +x) = Si+1 +α +(Sjk+1 +α +x) = Sjk+1+1 +α +x. +□ +Proposition 2.9. The matching parameters α ∈ [1, β] for Tα and for Sα coincide. +Proof. Recall that Tα has critical points at ±1/β, and note that limx→1/β− Tα(x) = 1 while limx→1/β+ Tα(x) = +β(1 − α). +Due to symmetry, Tα has matching if and only if there are integers M, N > 0 for which +T M +α (1) = T N +α (β(1 − α)). +Suppose first that Tα has matching. Then T M +α (1) = T N +α (β(1 − α)) for some M, N > 0. By (2) and the +fact that Sα(1−α) = β(1−α), this implies the existence of some M ′, N ′ > 0 for which SM ′ +α (1) = SN ′ +α (1−α). +Conversely, suppose Sα has matching with matching index m := m(α). From the proof of Proposition 2.1 +it is clear that Sm +α (1) = Sm +α (1 − α) ∈ J0. By Lemma 2.8, there are M, N > 0 for which +T M +α (1) = Sm+1 +α +(1) = Sm+1 +α +(1 − α) = Sm +α (β(1 − α)) = T N +α (β(1 − α)). +□ +2.2. Matching words and intervals. When Sα has matching, we call the first m(α) < ∞ digits of the +Sα-expansion of 1 the matching word corresponding to α. A maximal subinterval of [1, β] on which matching +words coincide is called a matching interval corresponding to the common matching word. Here we classify +matching words and matching intervals (Corollary 2.20); as all matching parameters belong to some matching +interval, this gives a complete classification of matching parameters α ∈ [1, β]. (Propositions 2.13, 2.18 and +2.19 imply that this also classifies the first m(α) < ∞ digits of the Sα-expansions of 1 − α for Sα with +matching and the maximal subintervals of parameters α on which these digits coincide.) Note that matching +words and intervals for α ∈ [1, β]\(1, 1 + 1/β3) have been implicitly determined via the cases considered in +§2.1. For instance, (1 + 1/β2, β] is the matching interval corresponding to the matching word 10, and the +Sα-expansion of 1 − α for each α ∈ (1 + 1/β2, β] begins with 0(−1). Similarly, (1 + 1/β3, 1 + 1/β2) is the +matching interval corresponding to the matching word 1001, and the Sα-expansion of 1 − α for each α in +this interval begins with 00(−1)0. +Denote by ≺ the lexicographical ordering on {0, ±1}N. +Note that ≺ may also be defined on the set +{0, ±1}∗ of finite words with alphabet −1, 0, 1 by first sending w ∈ {0, ±1}∗ to w0∞. +Definition 2.2. Let +w0 := 00 ≺ w1 := 001 ≺ w2 := 01. +We say that d ∈ {0, 1}∗ is in admissible block form if d = 10 or +d = 1wi1wi2 · · · win(1 − in/2) +for some i1, . . . , in ∈ {0, 1, 2}, n ≥ 1 with in ̸= 1, and, when n ≥ 2, i1 = 2. The collection of all words in +admissible block form is denoted B. +The condition that a word in admissible block form ends in win(1 − in/2), in ̸= 1, guarantees that the +final three digits are either 001 or 010 (recall (5)); however, not every word ending this way belongs to B: +Example 2.10. One verifies that +d := 1w2w0w1w01 = 10100001001 ∈ B, +whereas +d′ := 1010001 /∈ B. +9 + +Note that the indices ij for d ∈ B are uniquely determined; that is, if +1wi1wi2 · · · win(1 − in/2) = 1wj1wj2 · · · wjm(1 − jm/2), +then m = n and ik = jk for each 1 ≤ k ≤ n. Define ϕ : B → {0, −1}∗ by ϕ(10) = 01 and for each d ∈ B of +the form +d = 1wi1wi2 · · · win(1 − in/2), +by +ϕ(d) := 0w2−i1w2−i2 · · · w2−in(in/2), +where w := −w for each w ∈ {0, ±1}∗. +Let σ : {0, ±1}N → {0, ±1}N denote the left shift defined by σ((wj)j≥1) = (wj+1)j≥1 for each (wj)j≥1 ∈ +{0, ±1}N; as with the lexicographical ordering, σ is also defined on the set {0, ±1}∗ of finite words by sending +w ∈ {0, ±1}∗ to w0∞. We remark that for each T ∈ {Sα, Tα, B}, the left shift of the T-expansion of x +equals the T-expansion of T(x). +Definition 2.3. A word d ∈ B satisfies Property M if, for each j ≥ 0, both σj(d) ⪯ d and σj(ϕ(d)) ⪯ d. +Denote by M ⊂ B the collection of all words d satisfying Property M. We call 10 and 1001 the exceptional +words in M and denote by MU := M\{10, 1001} the collection of unexceptional words in M. +Example 2.11. Let d ∈ B be as in Example 2.10. Then +ϕ(d) = 0w0w2w1w20 = 00001001010, +and since both σj(d) ⪯ d and σj(ϕ(d)) ⪯ d for all j ≥ 0, we have d ∈ M. +We shall see that Property M classifies matching words of the maps Sα. To show that M contains all +matching words we need the following observation, which is not novel, but for which a proof is included for +completeness: +Lemma 2.12. Fix α ∈ [1, β] and x, y ∈ [−1, 1]. Then x < y if and only if (sα,j(x))j≥1 ≺ (sα,j(y))j≥1. +Similarly, for x, y ∈ [0, 1], x < y if and only if (bj(x))j≥1 ≺ (bj(y))j≥1. +Proof. Suppose x, y ∈ [−1, 1] with x < y, and let n := minj≥1{sα,j(x) ̸= sα,j(y)}. We first claim for each +0 ≤ j < n that Sj +α(x) < Sj +α(y). This is true by assumption for j = 0. If n = 1, we’re finished. Assume n > 1 +and that the claim holds for some j = k − 1 with 0 ≤ k − 1 < n − 1. Since sα,k(x) = sα,k(y), we have that +Sα restricts to a linear function with positive slope on an interval containing Sk−1 +α +(x) and Sk−1 +α +(y). But +Sk−1 +α +(x) < Sk−1 +α +(y) by assumption, so also Sk +α(x) < Sk +α(y) and the claim holds. Since sα,n(x) ̸= sα,n(y) and +Sn−1 +α +(x) < Sn−1 +α +(y), it must be true that sα,n(x) < sα,n(y) and hence (sα,j(x))j≥1 ≺ (sα,j(y))j≥1. +Now suppose x ≥ y. If equality holds, then by uniqueness of Sα-expansions, (sα,j(x))j≥1 = (sα,j(y))j≥1. +If the inequality is strict, the argument above applies with x and y interchanged. +The proof of the second statement is identical, mutatis mutandis. +□ +Proposition 2.13. Suppose for some α ∈ [1, β] that Sα has matching with index m := m(α), and let +d := d1 · · · dm denote the corresponding matching word. Then d ∈ M, and e := ϕ(d) agrees with the first m +digits e1 · · · em of the Sα-expansion of 1 − α. +Proof. From the cases of §2.1, the result holds for α /∈ (1, 1 + 1/β3); in particular, α ∈ (1 + 1/β2, β] and +α ∈ (1 + 1/β3, 1 + 1/β2) correspond to the exceptional words d = 10 and d = 1001, respectively, in M, and +ϕ(10) = 01, ϕ(1001) = 0010. Now assume α ∈ (1, 1 + 1/β3). Note that d1 = 1, e1 = 0, and +Sα(1) − Sα(1 − α) = (β − α) − β(1 − α) = α/β. +Recall from Equation (5) and the discussion preceding it that +�dm−2dm−1dm +em−2em−1em +� +∈ +��001 +010 +� +, +�010 +001 +�� += +��w01 +w20 +� +, +�w20 +w01 +�� +, +and Sm−3 +α +(1) − Sm−3 +α +(1 − α) = α/β. The remaining digits +�d2d3 · · · dm−3 +e2e3 · · · em−3 +� +10 + +are thus determined by edge labels of cycles in the graph of Figure 2 beginning and ending at vertex α/β. +There are three possible cycles, whose edge labels give +�djdj+1 +ejej+1 +� += +�01 +00 +� += +�w2 +w0 +� +, +�djdj+1 +ejej+1 +� += +�00 +01 +� += +�w0 +w2 +� +, and +�djdj+1dj+2 +ejej+1ej+2 +� += +�001 +001 +� += +�w1 +w1 +� +. +It follows that d = 1wi1wi2 · · · win(1−in/2) and e1 · · · em = 0w2−i1w2−i2 · · · w2−in(in/2) for some i1, . . . , in ∈ +{0, 1, 2}, n ≥ 1 and in ̸= 1. Moreover, note from case (v) of §2.1 that d1d2d3d4 = 1010, so i1 = 2. Thus +d ∈ B and e = e1 · · · em = ϕ(d). From Lemma 2.12, the facts that Sj +α(1), Sj +α(1 − α) ∈ [−1, 1] for each j ≥ 0 +imply that σj(d), σj(e) ⪯ d for each j ≥ 0. Thus d ∈ M. +□ +The previous result states that every matching word belongs to M. Before proving the converse (Propo- +sitions 2.16 and 2.18), we define and investigate properties of the valuation function v : S → R given by the +(absolutely) convergent series +v((wj)j≥1) := +� +j≥1 +wj/βj, +where S ⊂ ZN consists of all sequences (wj)j≥1 whose entries are bounded above and below. The valuation +function is also defined on the set S∗ ⊂ S of finite words by considering the corresponding finite sum and +setting v(ε) = 0 for the empty word ε. It is not difficult to check for finite words w, w′ ∈ {0, ±1}∗ with no +consecutive nonzero digits that w ≺ w′ if and only if v(w) < v(w′). +Lemma 2.14. If w := w1w2 · · · wk ∈ {0, 1, 2}∗ is ε (in which case we set k = 0) or consists solely of blocks +of 01’s and 002’s, then +v(w) = 1/β − 1/βk+1. +Proof. The case that w = ε is trivial, so suppose w ̸= ε. One easily verifies that +v((01)3) = v((002)2) +and +v(01002) = v(00201). +These observations, together with the fact that for each 1 ≤ j ≤ k, +v(w) = v(w1 · · · wj) + (1/βj)v(wj+1 · · · wk), +imply that +v(w) = +� +� +� +� +� +v((002)k/3), +k ≡ 0 (mod 3) +v((002)(k−4)/3(01)2), +k ≡ 1 (mod 3) +v((002)(k−2)/301), +k ≡ 2 (mod 3) +. +Notice that for any j ≥ 1, +v((002)j) = 2 +j +� +i=1 +(1/β3)i += 2 · 1/β3 − 1/β3j+3 +1 − 1/β3 += 2 · 1 − 1/β3j +β3 − 1 += 2 · 1 − 1/β3j +2β += 1/β − 1/β3j+1. +11 + +If k ≡ 0 (mod 3), setting j = k/3 gives the result. If k ≡ 1 (mod 3), we compute +v(w) = v((002)(k−4)/3(01)2) += v((002)(k−4)/3) + (1/βk−4)v((01)2) += 1/β − 1/βk−3 + (1/βk−4)(1/β2 + 1/β4) += 1/β − 1/βk−3 + 1/βk−2 + 1/βk += 1/β − 1/βk+1. +Similarly, if k ≡ 2 (mod 3), +v(w) = v((002)(k−2)/301) += v((002)(k−2)/3) + (1/βk−2)v(01) += 1/β − 1/βk−1 + 1/βk += 1/β − 1/βk+1. +□ +For equal-length words x, y ∈ {0, ±1}∗, define x+y, x−y ∈ {0, ±1, ±2}∗ where addition and subtraction, +respectively, are performed entry-wise. Note that +w2 − w0 = 01, +w0 − w2 = 01, +and +w1 − w1 = 002. +Suppose d satisfies Property M with m := len(d). Since d is in admissible block form, the definition of +e := ϕ(d) implies that d − e = 1w1 for some word w consisting solely of blocks of 01’s and 002’s or w = ε. +Using Lemma 2.14, we compute +v(d − e) = v(1w1) = 1/β + (1/β)(1/β − 1/βm−1) + 1/βm = 1. +This proves the following: +Proposition 2.15. If d ∈ M and e := ϕ(d), then +v(d) − v(e) = v(d − e) = 1. +For d = 10, set Id = (α− +d , α+ +d ] := (1 + 1/β2, β], and for all other d = d1 · · · dm ∈ M, define +Id = (α− +d , α+ +d ) := +� +βm + βdm +βmv(d) + βdm , +βm − β1−dm +βmv(d) − β1−dm +� +. +(7) +Proposition 2.16. For each d ∈ M, Id is a nonempty subinterval of (1, β]. +Proof. The result is true for d = 10, so assume d ̸= 10. We first show that Id ̸= ∅, i.e. that +βm + βdm +βmv(d) + βdm < +βm − β1−dm +βmv(d) − β1−dm , +or +(βm + βdm)(βmv(d) − β1−dm) < (βm − β1−dm)(βmv(d) + βdm). +Distributing and cancelling terms gives that this is equivalent to +βm+dmv(d) − βm+1−dm < βm+dm − βm+1−dmv(d), +or v(d) < 1. Since d has no consecutive 1’s, one finds that v(d) < v((10)∞) = 1 (see also Lemma 1 of [21]). +Next we show that Id ⊂ (1, β]. The left endpoint of Id is greater than 1 again since v(d) < 1. It remains +to show that +βm − β1−dm +βmv(d) − β1−dm ≤ β. +Recall that d1 = 1, and if dm = 0, then dm−1 = 1; thus v(d) ≥ 1/β + β1−dm/βm, and +βm+1v(d) − β2−dm ≥ βm+1(1/β + β1−dm/βm) − β2−dm > βm − β1−dm. +Dividing both sides by βmv(d) − β1−dm gives the desired inequality. +□ +12 + +For each u ∈ {0, 1}∗, let ∆(u) denote the cylinder of points x ∈ [0, 1] for which the β-expansion of x +begins with u. One finds for each u = u1 · · · un with ujuj+1 = 0, 1 ≤ j < n, that +∆(u) = +� +[v(u), v(u) + 1/βn), +un = 0 +[v(u), v(u) + 1/βn+1), +un = 1 . +(8) +The following lemma is needed in Proposition 2.18 below. +Lemma 2.17. Let d ∈ MU. Then Bj(1/α− +d ) ≤ 1/α− +d and Bj(1 − 1/α+ +d ) ≤ 1/α+ +d for all j > 0. +Proof. This is a corollary of two technical results (Lemmas 4.1 and 4.2), whose statements and proofs are +provided in the appendix. +□ +The next result—together with Proposition 2.16—states that every word d ∈ M is in fact a matching +word, thus completing our classification of matching words as the set M. Moreover, it states that the interval +Id is contained in a matching interval corresponding to the matching word d. +Proposition 2.18. For any d ∈ M and α ∈ Id, the Sα-expansions of 1 and 1 − α begin with d and ϕ(d), +respectively. Moreover, Sα has matching with matching index m(α) = len(d). +Proof. The result is shown for exceptional words d ∈ {10, 1001} in §2.1, so assume d ∈ MU. Suppose the +first statement holds. That Sα has matching with index m(α) = len(d) is implied by the final three digits +of d and e (see the discussion surrounding Equation (5)), so we need only prove the first statement. Let +α ∈ Id, and write d = d1 · · · dm and e := ϕ(d) = e1 · · · em. We must show that +dα,1 · · · dα,m = d1 · · · dm +and +eα,1 · · · eα,m = e1 · · · em. +Assume that +�dm−2dm−1dm +em−2em−1em +� += +�001 +010 +� +, +and set α0 := 1/v(d) (the case that dm = 0 is similar). Proposition 2.16 together with the fact that v(d) < 1 +imply α− < α0 < α+, where, for ease of notation, α± := α± +d . We claim that it suffices to show the following: +(i) if α ∈ (α−, α0), then ℓα(1) > m − 1, ℓα(1 − α) = m − 2, +b1(1/α) · · · bm(1/α) = d1 · · · dm, +and +b1(1 − 1/α) · · · bm−2(1 − 1/α) = e1 · · · em−2; +(ii) if α ∈ (α0, α+), then ℓα(1) = m − 1, ℓα(1 − α) > m − 2, +b1(1/α) · · · bm−1(1/α) = d1 · · · dm−1, +and +b1(1 − 1/α) · · · bm(1 − 1/α) = e1 · · · em; +and +(iii) if α = α0, then ℓα(1) = m − 1, ℓα(1 − α) = m − 2, +b1(1/α) · · · bm−1(1/α) = d1 · · · dm−1, +b1(1 − 1/α) · · · bm−2(1 − 1/α) = e1 · · · em−2, +and Bm−1(1/α) = Bm−2(1 − 1/α) = 1/β. +Indeed, suppose (i) holds. Lemma 2.5 implies +dα,1 · · · dα,m = d1 · · · dm +and +eα,1 · · · eα,m−2 = e1 · · · em−2. +Since ℓα(1 − α) = m − 2, Corollary 2.4 gives Sm−2 +α +(1 − α) ∈ [−α/β, −1/β), so eα,m−1 = −1 and eα,m = 0. +In case (ii), Lemma 2.5 again gives +dα,1 · · · dα,m−1 = d1 · · · dm−1 +13 + +and +eα,1 · · · eα,m−1 = e1 · · · em−1. +Moreover, ℓα(1) = m − 1 implies Sm−1 +α +(1) ∈ (1/β, α, β] and hence dα,m = 1. Since eα,m−1 = em−1 = −1, it +follows that eα,m = 0 = em. In (iii), we have +dα,1 · · · dα,m−1 = d1 · · · dm−1 +and +eα,1 · · · eα,m−2 = e1 · · · em−2. +Moreover, Lemma 2.5 gives Sm−1 +α +(1) = −Sm−2 +α +(1 − α) = α/β, so dα,m = eα,m−1 = 1 and eα,m = 0. +By Corollary 2.6, (i), (ii) and (iii) are implied by showing: +(a) 1/Id ⊊ ∆(d1 · · · dm−1) and 1 − 1/Id ⊊ ∆(e1 · · · em−2); +(b) Bj(1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m−1, and Bj(1−1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m−2; +(c) if α ∈ (α−, α0), then Bm−1(1/α) > 1/β and Bm−2(1 − 1/α) ∈ (1/βα, 1/β]; +(d) if α ∈ (α0, α+), then Bm−1(1/α) ∈ (1/βα, 1/β] and Bm−2(1 − 1/α) > 1/β; and +(e) if α = α0, then Bm−1(1/α) = Bm−2(1 − 1/α) = 1/β. +We prove each of (a), (b), (c), (d) and (e): +(a) The first inclusion is equivalent to +v(d1 · · · dm−1) < 1/α+ < 1/α− < v(d1 · · · dm−1) + 1/βm−1. +(9) +Note that v(d1 · · · dm−1) < 1/α+ if and only if +v(d) − 1/βm < βmv(d) − 1 +βm − 1 +. +Multiplying both sides by βm−1, cancelling and rearranging terms, this is equivalent to v(d) > 1/βm. +This latter inequality holds since v(d) ≥ v(d1) = 1/β and m > 1. Next, 1/α− < v(d1 · · · dm−1) + +1/βm−1 if and only if +βmv(d) + β +βm + β +< v(d) − 1/βm + 1/βm−1. +Using the fact that 1/βm−1 = 1/βm+1/βm+1 and multiplying both sides by βm+β, this is equivalent +to +βmv(d) + β < (βm + β)(v(d) + 1/βm+1), +or +βmv(d) + β < βmv(d) + 1/β + βv(d) + 1/βm. +Simplifying, this is equivalent to showing 1 < βv(d) + 1/βm, which again holds since v(d) ≥ 1/β. +Thus 1/Id ⊊ ∆(d1 · · · dm−1). +The second inclusion is equivalent to +v(e1 · · · em−2) < 1 − 1/α− < 1 − 1/α+ < v(e1 · · · em−2) + 1/βm−2. +Now v(e1 · · · em−2) < 1 − 1/α− if and only if 1/α− < 1 − (v(e) − 1/βm−1). By Proposition 2.15, the +fact that v(e) = −v(e) and (9), +1 − (v(e) − 1/βm−1) = v(d) + 1/βm−1 > v(d1 · · · dm−1) + 1/βm−1 > 1/α−. +Lastly, 1 − 1/α+ < v(e1 · · · em−2) + 1/βm−2 if and only if 1 − 1/α+ < v(e) − 1/βm−1 + 1/βm−2, or +v(d) < 1/α+ + 1/βm. From (9), we find +v(d) − 1/βm = v(d1 · · · dm−1) < 1/α+. +Thus 1 − 1/Id ⊊ ∆(e1 · · · em−2). +(b) Fix 0 ≤ j < m − 1. If dj+1 = 1, then part (a) and Lemma 2.12 imply that Bj(1/α) > Bj(1/α+) ≥ +1/β. +Now suppose dj+1 = 0. +By (a), Bj(1/α−) ∈ (1/βα−, 1/β] if and only if Bj+1(1/α−) ∈ +(1/α−, 1]. Lemma 2.17 thus implies Bj(1/α−) /∈ (1/βα−, 1/β]. By Equation (6), it also holds for +each x ∈ ∆(d1 · · · dm−1) that Bj(x) /∈ (x/β, 1/β] if and only if +βj(x − v(d1 · · · dj)) ≤ x/β, +14 + +or +x ≤ βjv(d1 · · · dj) +βj − 1/β +. +Since 1/α, 1/α− ∈ ∆(d1 · · · dm−1) and Bj(1/α−) /∈ (1/βα−, 1/β], we have +1/α < 1/α− ≤ βjv(d1 · · · dj) +βj − 1/β +, +which implies Bj(1/α) /∈ (1/βα, 1/β]. Thus Bj(1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m − 1. +The proof that Bj(1 − 1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m − 2 is similar. +(c) Suppose α ∈ (α−, α0). From Equation (6) and part (a), we have for each x ∈ 1/Id that +Bm−1(x) = βm−1(x − v(d1 · · · dm−1)) +(10) += βm−1(x − (v(d) − 1/βm)) +Since 1/α > 1/α0 = v(d), we have Bm−1(1/α) > 1/β. +Also from Equation (6), part (a) and +Proposition 2.15, for each x ∈ 1/Id, +Bm−2(1 − x) = βm−2(1 − x − v(e1 · · · em−2)) +(11) += βm−2(1 − x + v(e) + 1/βm−1) += βm−2(−x + v(d) + 1/βm−1) += −βm−2x + βm−2v(d) + 1/β. +Hence +Bm−2(1 − 1/α) < Bm−2(1 − 1/α0) = 1/β, +and Bm−2(1 − 1/α) > 1/βα if and only if +βm−2v(d) + 1/β +βm−2 + 1/β +> 1/α. +But the left hand side equals 1/α−, so the inequality holds. +(d) Suppose α ∈ (α0, α+). +From Equation (10), 1/α < 1/α0 = v(d) implies Bm−1(1/α) < 1/β. +Moreover, Bm−1(1/α) > 1/βα if and only if +1/α > βm−1v(d) − 1/β +βm−1 − 1/β +. +The right-hand side equals 1/α+, and α < α+ by assumption. We also find from Equation (11) that +Bm−2(1 − 1/α) = βm−2(v(d) − 1/α) + 1/β > 1/β +since 1/α < 1/α0 = v(d). +(e) This again follows from Equations (10) and (11), setting x = 1/α0 = v(d). +□ +The following proposition states that the interval Id contains the matching intervals corresponding to +the matching word d; together with Proposition 2.18, this characterises matching intervals as the collection +{Id}d∈M. +Proposition 2.19. If Sα has matching with m(α) = m, then α ∈ Id, where d = d1 · · · dm is beginning of +the Sα-expansion of 1. +Proof. By Proposition 2.13, d ∈ M, so Id is defined. The result holds for m ≤ 2 by the cases in §2.1, so +assume m > 2 and let e = e1 · · · em denote the beginning of the Sα-expansion of 1−α. Recall from Equation +(5) that +�dm−2dm−1dm +em−2em−1em +� +∈ +��010 +001 +� +, +�001 +010 +�� +. +Assume dm = 0 (the other case is similar). Lemma 2.3, Corollary 2.4 and the final digits of d and e imply +that either +(i) Sm−2 +α +(1) ∈ (1/β, α/β] +or +(ii) Sm−1 +α +(1 − α) ∈ [−α/β, −1/β). +15 + +It suffices to show that both (i) and (ii) imply +α ∈ Id = +� +βm + 1 +βmv(d) + 1, +βm − β +βmv(d) − β +� +. +(i) Equation (3) gives +Sm−2 +α +(1) = βm−2(1 − αv(d1 · · · dm−2)) ∈ (1/β, α/β]. +Note that v(d1 · · · dm−2) = v(d) − 1/βm−1, so +1 − α(v(d) − 1/βm−1) ∈ (1/βm−1, α/βm−1]. +Now +1 − α(v(d) − 1/βm−1) > 1/βm−1 +implies +α < +1 − 1/βm−1 +v(d) − 1/βm−1 = +βm − β +βmv(d) − β . +Moreover, +1 − α(v(d) − 1/βm−1) ≤ α/βm−1 +gives 1 ≤ αv(d). Thus we have +α ∈ +� +1 +v(d), +βm − β +βmv(d) − β +� +, +and it suffices to show that +βm + 1 +βmv(d) + 1 < +1 +v(d). +But this is true since v(d) < v((10)∞) = 1. +(ii) Again from Equation (3), +Sm−1 +α +(1 − α) = βm−1(1 − α(1 + v(e1 · · · em−1))) ∈ [−α/β, −1/β). +The assumption that em = −1 together with Proposition 2.15 give +1 + v(e1 · · · em−1) = 1 + v(e) + 1/βm = v(d) + 1/βm, +so +1 − α(v(d) + 1/βm) ∈ [−α/βm, −1/βm). +Now +1 − α(v(d) + 1/βm) ≥ −α/βm +implies 1 ≥ αv(d). Furthermore, +1 − α(v(d) + 1/βm) < −1/βm +gives +α > +1 + 1/βm +v(d) + 1/βm = +βm + 1 +βmv(d) + 1. +Hence +α ∈ +� +βm + 1 +βmv(d) + 1, +1 +v(d) +� +, +and it suffices to show +1 +v(d) < +βm − β +βmv(d) − β . +This is true again since v(d) < 1. +□ +The implications of Propositions 2.13, 2.16, 2.18 and 2.19 are summarised in the following: +Corollary 2.20. The sets M and {Id}d∈M classify the matching words and intervals, respectively, of the +maps Sα. +16 + +Remark 2.21. The results of this subsection also imply that ϕ(M) classifies the first m(α) < ∞ digits of +the Sα-expansions of 1 − α for matching parameters α ∈ [1, β]. Moreover, the intervals Id in {Id}d∈M = +{Iϕ−1(e)}e∈ϕ(M) classify the maximal subintervals of matching parameters α for which these first m(α) digits +coincide (and equal e = ϕ(d)). +Remark 2.22. While not needed for our purposes, we briefly mention that the sets M (or ϕ(M)) and +{Id}d∈M also give rise to classifications of the Tα-expansions of 1 (resp. β(1 − α)) before matching and +the maximal intervals of parameters α on which these expansions coincide. In particular, if d ∈ M (resp. +e := ϕ(d) ∈ ϕ(M)), then the corresponding Tα-word d′ (resp. +e′) ‘forgets’ each non-terminal 0 which +immediately follows a 1 (resp. −1, and e′ also forgets the initial 0 of e). The matching intervals Id are +unchanged. For instance, d = 10100001 and e = ϕ(d) = 00001010 give rise to the words d′ = 110001 and +e′ = 000110 for Tα, and each of these words corresponds to the matching interval Id = +� +β8+β +β7+β5+β2 , β8−1 +β7+β5 +� +. +2.3. Cascades of matching intervals. Here it is shown that each unexceptional matching interval Id, d ∈ +MU, generates a whole ‘cascade’ of unexceptional matching intervals with adjacent endpoints. Define ψ : +MU → {0, 1}∗, where for d = d1 · · · dm ∈ MU and e := ϕ(d) = e1 · · · em, +ψ(d) = +� +de, +dm = 0 +de2 · · · em, +dm = 1 . +Recall the definition of the matching interval Id = (α− +d , α+ +d ) from (7). +Proposition 2.23. The map ψ preserves Property M, i.e. ψ(MU) ⊂ MU. Moreover, α− +d = α+ +ψ(d) for each +d ∈ M. +Proof. Let d = d1 · · · dm ∈ MU, and assume dm = 0 (the other case is similar). We first show α− +d = α+ +ψ(d), +assuming ψ(MU) ⊂ MU. We compute +α+ +ψ(d) = +β2m − 1 +β2mv(de) − 1 += +(βm + 1)(βm − 1) +β2m(v(d) − (1/βm)v(e)) − 1 += +(βm + 1)(βm − 1) +β2mv(d) − βm(v(d) − 1) − 1 += +(βm + 1)(βm − 1) +(βmv(d) + 1)(βm − 1) += +βm + 1 +βmv(d) + 1 += α− +d +as desired. Now we prove that d′ := ψ(d) ∈ MU. Clearly d′ /∈ {10, 1001}, so we need only show d′ ∈ M. +Write +d = 1wi1 · · · win0 +with in = 2 and +e = ϕ(d) = 0w2−i1 · · · w2−in1. +Then +d′ = de += 1wi1 · · · win00w2−i1 · · · w2−in1 += 1wi1 · · · winw0w2−i1 · · · w2−in1, +so d′ ∈ B is in admissible block form. To prove d′ ∈ M, it remains to show for each j ≥ 0 that (i) σj(d′) ⪯ d′ +and (ii) σj(ϕ(d′)) ⪯ d′. (Recall that d ∈ M implies the analogous inequalities hold for d.) +17 + +(i) If j ≥ m, then +σj(d′) = σj(de) = σj−m(e) ⪯ d ⪯ d′. +Assume j < m, and suppose for the sake of contradiction that σj(d′) ≻ d′. Since d′ begins with 1, +so does σj(d′). Thus either +σj(d′) = 1wiℓ · · · winw0w2−i1 · · · w2−in1 +for some 1 < ℓ ≤ n, or +σj(d′) = 1w0w2−i1 · · · w2−in1. +Since w0 ≺ w2 = wi1, the second case is impossible and we must have +1wiℓ · · · winw0w2−i1 · · · w2−in1 ≻ 1wi1 · · · winw0w2−i1 · · · w2−in1 +for some ℓ. Since σj(d) ⪯ d, it follows that +1wiℓ · · · win = 1wi1 · · · win−ℓ+1 +and thus +w0w2−i1 · · · w2−in1 ≻ win−ℓ+2 · · · winw0w2−i1 · · · w2−in1. +Then either there is some 1 ≤ p ≤ ℓ − 3 for which +(0, 2 − i1, . . . , 2 − ip−1) = (in−ℓ+2, in−ℓ+3, . . . , in+p−ℓ+1) +and 2 − ip > in+p−ℓ+2, or +(0, 2 − i1, . . . , 2 − iℓ−2) = (in−ℓ+2, in−ℓ+3, . . . , in). +In the first case, +(2 − in−ℓ+2, 2 − in−ℓ+3, . . . , 2 − in+p−ℓ+1) = (2, i1, . . . , ip−1) +and 2 − in+p−ℓ+2 > ip. Thus there exists some k ≥ 0 for which +σk(e) = 1w2−in−ℓ+3 · · · w2−in+p−ℓ+1w2−in+p−ℓ+2 · · · w2−in1 +≻ 1wi1 · · · wip−1wip · · · win0 += d, +contradicting the fact that d ∈ M. In the second case, +(2 − in−ℓ+2, 2 − in−ℓ+3, . . . , 2 − in) = (2, i1, . . . , iℓ−2). +Since in = 2 implies iℓ−2 = 0, there is again some k ≥ 0 for which +σk(e) = 1w2−in−ℓ+3 · · · w2−in−1w2−in1 += 1w2−in−ℓ+3 · · · w2−in−1w1 +≻ 1wi1 · · · wiℓ−3wiℓ−2 · · · win0 += d, +contradicting d ∈ M. +(ii) Set e′ := ϕ(d′) = e1 · · · em, and recall that dm = 0 implies em = −1. Then +e′ = 0w2−i1 · · · w2−inw2wi1 · · · win0 += e1 · · · em−10d. +If j < m − 1, then +σj(e′) = ej+1 · · · em−10d ≺ ej+1 · · · em = σj(e) ⪯ d ⪯ d′. +If j = m − 1, then +σj(e′) = 0d ≺ d′, +and if j ≥ m, then +σj(e′) = σj−m(d) ⪯ d ⪯ d′. +This concludes the proof that d′ = ψ(d) ∈ M and thus ψ(MU) ⊂ MU. +□ +18 + +3. Invariant measures and frequencies of digits +As noted above, our main interest in matching arises from results of [17] which provide explicit expressions +for the densities of absolutely continuous invariant measures. These densities depend on the orbits of the +left and right limits at critical points and are in general infinite sums of (finite) step functions; however, the +infinite sum becomes finite when either matching or a Markov partition occurs. These observations are used +in this section to obtain explicit invariant measures να and µα for the maps Sα and Tα, respectively, and +asymptotic relative frequencies of digits occurring in their respective generic expansions. These measures +and frequencies are used in the proofs of Theorems 1.1 and 1.2. +Recall that B(x) := βx (mod 1). It is well known that +h(x) := +� +5+3 +√ +5 +10 +, +x ∈ [0, 1/β) +5+ +√ +5 +10 , +x ∈ [1/β, 1] +is the density of a unique, ergodic, B-invariant probability measure which is equivalent to Lebesgue measure +λ ([22]). By Birkhoff’s ergodic theorem, the frequency of 0 in λ-a.e. β-expansion is +� +[0,1/β) hdλ = (5+ +√ +5)/10. +When α = 1, the map Sα = S1 restricts on [0, 1]\{1/β} to B and on [−1, 0]\{−1/β} to −B(−x). Since S1 +is invariant on ±[0, 1], we find that the frequency of 0 in λ-a.e. S1-expansion is also (5 + +√ +5)/10. Define f1 : +[−1, 1] → [−1, 1] by f1(x) = h(|x|)/2, and recall the definitions of the subintervals Ji ⊂ [−1, 1], i ∈ {−1, 0, 1} +from §1. Note, then, that the measure ν1 defined on Lebesgue-measurable A ⊂ [−1, 1] by ν1(A) = +� +A f1dλ +satisfies ν1(J0) := (5 + +√ +5)/10. +A similar analysis (with Lebesgue measure) reveals that the frequency of 0 in λ-a.e. T1-expansion is 1/β. +Setting µ1 := λ/2 as normalised Lebesgue measure gives µ1(J0) = 1/β. In what follows we consider α ̸= 1. +3.1. Invariant measures. Let α ∈ (1, β]. Following a procedure completely analogous to that in §2.1 of +[13], results of [17] imply that the collection of absolutely continuous Sα-invariant measures forms a one real- +dimensional linear space and thus there is a unique—and hence ergodic—absolutely continuous invariant +probability measure να. Moreover, its corresponding probability density is given explicitly by +fα(x) := 1 +C +� +t≥0 +1 +βt+1 +� +1[−1,St +α(α−1))(x) − 1[−1,St +α(−1))(x) + 1[−1,St +α(1))(x) − 1[−1,St +α(1−α))(x) +� +, +where C ∈ R is some normalising constant. Symmetry of Sα together with Proposition 2.1 allow us to +rewrite fα(x) as +fα(x) = 1 +C +� +t≥0 +1 +βt+1 +� +1[Stα(−1),Stα(α−1))(x) + 1[Stα(1−α),Stα(1))(x) +� +. +(12) +Note that fα is bounded away from 0 on [−1, 1), so να is in fact equivalent to Lebesgue measure λ. Also +observe that when matching (or a Markov partition) occurs, the summation becomes a finite sum and fα(x) +is a (finite) step function (see Figure 3). +The measure να can now be used to obtain a unique, absolutely continuous Tα-invariant measure µα = +� +gαdλ. For each α ∈ (1, β], define a probability measure +µα(A) := να +� +S−1 +α (A) ∩ J0 +� +να(J0) +. +(13) +on [−1, 1], where A ⊂ [−1, 1] is Lebesgue-measurable. Note that S−1 +α (A) ∩ J0 = +1 +β A, so µα may also be +written µα(A) = να( 1 +β A)/να(J0). +Theorem 3.1. The measure µα is the unique—hence ergodic—invariant probability measure for Tα which +is absolutely continuous with respect to Lebesgue measure. Moreover, µα is equivalent to Lebesgue measure. +Proof. Since Tα is an expanding, piecewise C2 monotone map, results of [19] imply the existence of an +invariant probability measure ρα for Tα which is absolutely continuous with respect to Lebesgue measure. +Let J±1 := J−1∪J1. As Tα is a jump transformation for Sα, the measure ρα induces an Sα-invariant measure +defined by +˜ρα(A) := ρα(A) + ρα +� +S−1 +α (A) ∩ J±1 +� +(14) +19 + +Figure 3. The invariant densities fα for Sα (red) and gα for Tα (blue) with α = 1.16 (left), +α = 1/v(1010) ≈ 1.17082 . . . (center) and α = 1.2 (right). +(see, e.g. Proposition 11.4.1 of [14]). Note that for any A ⊂ J±1 we have S−1 +α (A) ⊂ J0, so (14) gives +˜ρα(A) = ρα(A). Then for any measurable A ⊂ [−1, 1], +˜ρα +� +S−1 +α (A) ∩ J±1 +� += ρα +� +S−1 +α (A) ∩ J±1 +� +and (14) gives +ρα(A) = ˜ρα(A) − ˜ρα +� +S−1 +α (A) ∩ J±1 +� +. +Since ˜ρα is Sα-invariant, the previous line may be rewritten +ρα(A) = ˜ρα(S−1 +α (A)) − ˜ρα +� +S−1 +α (A) ∩ J±1 +� += ˜ρα(S−1 +α (A) ∩ J0). +Recall that να is the unique invariant, absolutely continuous probability measure for Sα, so ˜ρα = cνα for +some c > 0. Thus +ρα(A) = cνα +� +S−1 +α (A) ∩ J0 +� +, +and setting A = [−1, 1] gives c = 1/να(J0). Hence ρα = µα. +That µα is equivalent to Lebesgue measure λ follows immediately from the fact that να is equivalent to +λ and the observation above that µα(A) = να( 1 +β A)/να(J0). +□ +We are now ready to prove Theorem 1.1: +Proof of Theorem 1.1. Theorem 3.1 asserts the existence of a unique, absolutely continuous Tα-invariant +probability measure µα which is in fact equivalent to Lebesgue measure. It remains to show that for fixed +d ∈ M, the density gα of each µα, α ∈ Id, is a step function with at most the same, finite number of jumps. +Using a change of variables, one finds that +µα(A) = +να( 1 +β A) +να(J0) = +1 +να(J0) +� +1 +β A +fα(x)dλ(x) = +1 +βνα(J0) +� +A +fα(x/β)dλ(x), +so +gα(x) = fα(x/β) +βνα(J0) . +Since, by (12), fα is a linear combination of at most 2m(α) indicator functions and m(α) is constant on Id, +the result follows. +□ +Remark 3.2. The number of jumps of the invariant densities fα and gα for Sα and Tα, respectively, +are non-constant on matching intervals Id. +Figure 3 shows these densities for three values of α in the +matching interval Id ≈ (1.14589 . . . , 1.23606 . . . ) with d = 1010. Note that the number of jumps is fewer +for α = 1/v(d). One can show that this phenomenon generalises to all matching intervals; in fact, for each +d ∈ M, the number of jumps of fα and gα, respectively, are constant for all but finitely many α ∈ Id, and +the number of jumps decreases for α = 1/v(d) ∈ Id. +20 + +0.8 +0.6 - +0.4 - +0.2 +1.0 +0.5 +0.5 +1.00.8 +0.6 - +0.4 - +0.2 +-1.0 +0.5 +0.5 +1.00.8 +0.6- +0.4 - +0.2 +1.0 +0.5 +0.5 +1.0Figure 4. The frequency functions fS(α) (red) and fT (α) (blue) plotted on all matching +intervals Id with len(d) ≤ 20. The visible plateaux correspond to the interval [1/2+1/β, 1+ +1/β2]. +3.2. Frequencies of digits. We are now in a position to determine the frequencies of digits in generic Sα- +and Tα-expansions. Define fS, fT : [1, β] → [0, 1] by +fS(α) := να(J0) +and +fT (α) := µα(J0). +For α ̸= 1, Birkhoff’s ergodic theorem—together with the equivalence of the ergodic measures να and µα +with Lebesgue measure λ—implies that the asymptotic frequencies +lim +n→∞ +1 +n +n−1 +� +i=0 +1J0(Si +α(x)) +and +lim +n→∞ +1 +n +n−1 +� +i=0 +1J0(T i +α(x)) +of the digit 0 in Lebesgue-a.e. Sα- and Tα-expansion are given by fS(α) and fT (α), respectively. Indeed, with +the discussion and notation given at the beginning of §3, fS(1) and fT (1) also give the generic asymptotic +frequencies of the digit 0. Note, too, that the frequencies of the digits ±1 are readily obtained from the +frequency of 0. +As in the proof of Theorem 3.1, set J±1 := J−1 ∪ J1. Using (13) and the Sα-invariance of να, one has for +any measurable A ⊂ [−1, 1], +µα(A) = να(S−1 +α (A)) − να(S−1 +α (A) ∩ J±1) +να(J0) += να(A) − να(S−1 +α (A) ∩ J±1) +να(J0) +. +Setting A = J0 and using the fact that S−1 +α (J0) ∩ J±1 = J±1, we find +µα(J0) = να(J0) − να(J±1) +να(J0) += να(J0) − (1 − να(J0)) +να(J0) +or +fT (α) = 2 − +1 +fS(α). +(15) +Proposition 3.3. The frequency functions fS and fT are continuous. +Proof. Arguments completely analogous to those in §4 of [13] give that fS is continuous. Continuity of fT is +immediate from 15. +□ +The remainder of this subsection is devoted to finding—for matching parameters α—an explicit expression +for fS(α) in terms of α and its corresponding matching word d (see Figure 4). Density of matching parameters +in [1, β], continuity of fS and equation (15) then allow us to determine fS(α) and fT (α) for any α ∈ [1, β] +as limits of these explicit expressions. These expressions are then used in §3.3 to determine the maximal +21 + +0.80 +0.75 +0.70 +0.65 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6frequency of the digit 0 occurring in generic Sα- and Tα-expansions, and it is shown that these maximal +values are attained for α in the interval [1/2 + 1/β, 1 + 1/β2]. +Assume that α ∈ Id, d ∈ M, with matching index m := m(α) < ∞, and recall the density fα from +equation (12). We first find an expression for the normalising constant C. By symmetry of Sα, +1 = να([−1, 1]) += +� 1 +−1 +fα(x)dλ(x) += 2 +C +m−1 +� +t=0 +� 1 +−1 +1 +βt+1 1[Stα(1−α),Stα(1))(x)dλ(x) += 2 +C +m−1 +� +t=0 +1 +βt+1 +� +St +α(1) − St +α(1 − α) +� +. +Assume α < 1 + 1/β2 and write +d = d1 · · · dm = 1wi1 · · · win(1 − in/2). +For each i ∈ 0, 1, 2, let ℓ(i) ∈ {2, 3} denote the length of the block wi—explicitly, ℓ(0) = ℓ(2) = 2 and +ℓ(1) = 3—and let p := pd : {1, . . . , n} → {1, . . . , m−3} be defined by p(k) = 1+�k−1 +j=1 ℓ(ij) so that σp(k)(d) = +wik · · · win(1−in/2). Recall from Figure 2 that S0 +α(1)−S0 +α(1−α) = α, Sm−1 +α +(1)−Sm−1 +α +(1−α) = α/β, and +that the remaining differences St +α(1)−St +α(1−α) are determined by cycles of length two or three beginning at +vertex α/β. In particular, if ik ∈ {0, 2}, then Sp(k) +α +(1)−Sp(k) +α +(1−α) = α/β and Sp(k)+1 +α +(1)−Sp(k)+1 +α +(1−α) = α +give a cycle of length two, while if ik = 1, Sp(k) +α +(1) − Sp(k) +α +(1 − α) = α/β, Sp(k)+1 +α +(1) − Sp(k)+1 +α +(1 − α) = α +and Sp(k)+2 +α +(1) − Sp(k)+2 +α +(1 − α) = βα give a cycle of length three. We find for each k ∈ {1, . . . , n} that +p(k)+ℓ(ik)−1 +� +t=p(k) +1 +βt+1 +� +St +α(1) − St +α(1 − α) +� += +ℓ(ik) +βp(k)+2 α, +and thus +1 = 2 +C +m−1 +� +t=0 +1 +βt+1 +� +St +α(1) − St +α(1 − α) +� += 2 +C +� +�α +β + +n +� +k=1 +p(k)+ℓ(ik)−1 +� +t=p(k) +1 +βt+1 +� +St +α(1) − St +α(1 − α) +� ++ +α +βm+1 +� +� += 2α +C +� +1 +β + +n +� +k=1 +ℓ(ik) +βp(k)+2 + +1 +βm+1 +� +. +(16) +Note that (16) also holds for α > 1 + 1/β2 (i.e. d = 10) with the summation over k set to zero. Define +a substitution ξ : {w0, w1, w2} → {02, 030} by ξ(w0) = ξ(w2) = 02 and ξ(w1) = 030, and let Ξ : M → +{0, 1, 2, 3}∗ be given by Ξ(d) = 101 if d = 10, and +Ξ(d) = 1ξ(wi1) · · · ξ(win)01 +if d = 1wi1 · · · win(1 − in/2) ∈ M\{10}. +The left- and right-most sides of (16) may be written more +succinctly as 1 = 2α +C v(Ξ(d)), and thus C = 2αv(Ξ(d)). +22 + +Having found C, we are now in a position to determine fS(α). Again by symmetry of Sα, +fS(α) = να(J0) += 1 − να(J−1) − να(J1) += 1 − +� −1/β +−1 +fα(x)dλ(x) − +� 1 +1/β +fα(x)dλ(x) += 1 − 2 +C +m−1 +� +t=0 +�� −1/β +−1 +1 +βt+1 1[Stα(1−α),Stα(1))(x)dλ(x) + +� 1 +1/β +1 +βt+1 1[Stα(1−α),Stα(1))(x)dλ(x) +� +. +Write e := ϕ(d) = e1 · · · em. Since by Proposition 2.1, St +α(1) /∈ J−1 and St +α(1 − α) /∈ J1 for t < m, the +previous line may be rewritten as +fS(α) = 1 − 2 +C +� +� +� +� +� +0≤t≤m−1 +et+1=−1 +1 +βt+1 (−1/β − St +α(1 − α)) + +� +0≤t≤m−1 +dt+1=1 +1 +βt+1 (St +α(1) − 1/β) +� +� +� +� += 1 − 2 +C +� +� +� +� +� +0≤t≤m−1 +dt+1=1 +1 +βt+1 St +α(1) − +� +0≤t≤m−1 +et+1=−1 +1 +βt+1 St +α(1 − α) − 1/β +� +� +� +� , +where we have used Proposition 2.15 together with the facts that +� +0≤t≤m−1 +et+1=−1 +1/βt+1 = −v(e) +and +� +0≤t≤m−1 +dt+1=1 +1/βt+1 = v(d). +Let d0 +1 = e0 +1 = ε be the empty word, and for 1 ≤ t ≤ m − 1 set dt +1 := d1 · · · dt and et +1 := e1 · · · et. For each +0 ≤ t ≤ m − 1, equation (3) gives St +α(1) = βt(1 − αv(dt +1)) and St +α(1 − α) = βt(1 − α − αv(et +1)). Setting +n(d) := #{1 ≤ j ≤ m | dj = 1} − #{1 ≤ j ≤ m | ej = −1}, +(17) +the frequency function may be written as +fS(α) = 1 − 2 +C +� +� +� +� +� +0≤t≤m−1 +dt+1=1 +1 +βt+1 βt(1 − αv(dt +1)) − +� +0≤t≤m−1 +et+1=−1 +1 +βt+1 βt(1 − α − αv(et +1)) − 1/β +� +� +� +� += 1 − 2 +βC +� +� +� +� +� +0≤t≤m−1 +dt+1=1 +(1 − αv(dt +1)) − +� +0≤t≤m−1 +et+1=−1 +(1 − α − αv(et +1)) − 1 +� +� +� +� += 1 − 2 +βC +� +� +� +�n(d) − α +� +� +� +� +� +0≤t≤m−1 +dt+1=1 +v(dt +1) − +� +0≤t≤m−1 +et+1=−1 +(1 + v(et +1)) +� +� +� +� − 1 +� +� +� +� . +Letting +Kd := +� +0≤t≤m−1 +dt+1=1 +v(dt +1) − +� +0≤t≤m−1 +et+1=−1 +(1 + v(et +1)) +and recalling that C = 2αv(Ξ(d)), we find +fS(α) = 1 − +1 +βv(Ξ(d)) +�n(d) − 1 +α +− Kd +� +. +(18) +23 + +Example 3.4. Let d = 1001. Then e = 0010, so n(d) = 1. Moreover, +v(Ξ(d)) = v(10201) = 1 +β + 2 +β3 + 1 +β5 +and +Kd = v(ε) + v(100) − (1 − v(00)) = − 1 +β2 . +Thus for all α ∈ I1001, +fS(α) = 1 − +1 +β3(1/β + 2/β3 + 1/β5) = 4/5. +A similar calculation with d = 1010 reveals that fS(α) = 4/5 also for all α ∈ I1010. +Before turning toward the maximal frequency of the digit 0, we give an alternate expression for Kd which +will be helpful below. Note that the first summation in the definition of Kd may be rewritten as the sum +of all v(dt +1), 1 ≤ t ≤ m, for which dt=1, excluding the greatest such index t. The second sum may be +similarly rewritten (though an extra term 1 appears from the first non-zero summand of the original sum). +Now suppose d ̸= 10. Recalling that {dm−2dm−1dm, em−2em−1em} = {001, 010}, we have +Kd = +� +1≤t≤m−3 +dt=1 +v(dt +1) − +� +� +�1 + +� +1≤t≤m−3 +et=−1 +(1 + v(et +1)) +� +� +� += v(1) + +� +1≤k≤n−1 +ik∈{1,2} +v(1wi1 · · · wik) − +� +� +� +�1 + +� +1≤k≤n−1 +2−ik∈{1,2} +(1 − v(0w2−i1 · · · w2−ik)) +� +� +� +� . +Recall that p(k + 1), 1 ≤ k ≤ n − 1, gives the power for which σp(k+1)(d) = wik+1 · · · win(1 − in/2); in +particular, p(k + 1) equals the length of 1wi1 · · · wik. By Lemma 2.14, +v(1wi1 · · · wik) + v(0w2−i1 · · · w2−ik) = 1 +β + 1 +β +� 1 +β − +1 +βp(k+1) +� += 1 − 1/βp(k+1)+1. +Then +Kd = 1 +β + +� +1≤k≤n−1 +ik∈{1,2} +v(1wi1 · · · wik) − +� +� +� +�1 + +� +1≤k≤n−1 +2−ik∈{1,2} +� +v(1wi1 · · · wik) + 1/βp(k+1)+1� +� +� +� +� += − 1 +β2 + +� +1≤k≤n−1 +ik=2 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +ik=0 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +2−ik∈{1,2} +1/βp(k+1)+1. +The latter summation equals +� +1≤k≤n−1 +2−ik∈{1,2} +1/βp(k+1)+1 = 1 +β v(0w2−i1 · · · w2−in−1) += 1 +β +� +v(e) − +1 +βm−3 v(w2−inin/2) +� += 1 +β +� +1 − v(d) − +1 +βm−3 v(w2−inin/2) +� += 1 +β +� +1 − v(d1 · · · dm−3) − +1 +βm−3 v(011) +� += 1 +β − 1 +β v(d1 · · · dm−3) − +1 +βm−1 , +24 + +and thus for d ∈ M\{10}, +Kd = −1 + +� +1≤k≤n−1 +ik=2 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +ik=0 +v(1wi1 · · · wik) + 1 +β v(d1 · · · dm−3) + +1 +βm−1 . +(19) +3.3. Maximal frequency of zero. Here we prove that the frequency functions fS and fT attain their +maximums on the (maximal) interval [1/2 + 1/β, 1 + 1/β2]. We first need some preliminary results. Note +that by (18), on the matching interval Id the frequency function fS is strictly increasing with α for n(d) > 1, +strictly decreasing for n(d) < 1 and constant for n(d) = 1. By (15), the same monotonicity conditions hold +for fT . +The first of our preliminary results states that fS (and hence fT ) is constant on ‘cascade’ intervals: +Lemma 3.5. For each d ∈ MU, we have n(ψ(d)) = 1. In particular, for each d ∈ MU, the frequency +function fS is constant on [limn→∞ α− +ψn(d), α− +d ]. +Proof. It suffices to prove the first statement; the second follows immediately from this, Proposition 2.23 +and continuity of fS. Write +d = d1 · · · dm = 1wi1 · · · win(1 − in/2) +and +e := ϕ(d) = e1 · · · em = 0w2−i1 · · · w2−in(in/2). +Observe that +d′ := ψ(d) = +� +de, +dm = 0 +de2 · · · em, +dm = 1 += +� +1wi1 · · · win00w2−i1 · · · w2−in(in/2), +dm = 0 +1wi1 · · · win−1001w2−i1 · · · w2−in(in/2), +dm = 1 += +� +1wi1 · · · winw0w2−i1 · · · w2−in(in/2), +dm = 0 +1wi1 · · · win−1w1w2−i1 · · · w2−in(in/2), +dm = 1 , +so +e′ := ϕ(d′) = +� +0w2−i1 · · · w2−inw2wi1 · · · win(1 − in/2), +dm = 0 +0w2−i1 · · · w2−in−1w1wi1 · · · win(1 − in/2), +dm = 1 += +� +e1 · · · em−10d, +dm = 0 +e1 · · · em−20d, +dm = 1 . +Recall that if dm = 0, then em = 1. In this case d′ has exactly one more digit 1 than does e′. If dm = 1, +then em−1em = 10. Since e1 = 0, we see that in this case, too, d′ has exactly one more digit 1 than does e′. +Thus in both cases n(d′) = 1. +□ +We make note here of some computations which will be useful below. Let c, ℓ ∈ Z with ℓ ≥ 0: +v((0c)ℓ) = c +ℓ +� +j=1 +1/β2j = c +β2 · 1 − 1/β2ℓ +1 − 1/β2 = c +β (1 − 1/β2ℓ) +(20) +v((00c)ℓ) = c +ℓ +� +j=1 +1/β3j = c +β3 · 1 − 1/β3ℓ +1 − 1/β3 = c +2β (1 − 1/β3ℓ) +(21) +v((0c0)ℓ) = βv((00c)ℓ) = c +2(1 − 1/β3ℓ) +(22) +v((000c)ℓ) = c +ℓ +� +j=1 +1/β4j = c +β4 +1 − 1/β4ℓ +1 − 1/β4 = +c +β(β2 + 1)(1 − 1/β4ℓ) +(23) +v((0c00)ℓ) = β2v((000c)ℓ) = +cβ +β2 + 1(1 − 1/β4ℓ). +(24) +25 + +Lemma 3.6. If α ∈ Id for some d ∈ M with n(d) = 1, then fS(α) ≤ 4/5. Moreover, equality holds if and +only if d ≺ 1(w2w0)∞. +Proof. Note that n(10) = 0, so we may assume d ≻ 10. That fS(α) = 4/5 for all α ∈ I1010 ∪ I1001 was shown +in Example 3.4. Thus we may assume that d ≻ 1010. Write +d = d1 · · · dm = 1wi1 · · · win(1 − in/2) = 1X1Y1 · · · XtYtwin(1 − in/2), +where each Xs and Ys, 1 ≤ s ≤ t, consists solely of w2i’s and w1’s, respectively, and each Xs, Ys ̸= ε +except possibly Yt. Let ℓ2s−1 := 1 +2len(Xs) and ℓ2s := 1 +3len(Ys) denote the number of blocks wi in Xs and +Ys, respectively, and set ℓj := 0 for j > 2t. Analogous to the function p = pd defined in §3.2, set p1 := 1 +and for each s ≥ 1, let p2s := p2s−1 + 2ℓ2s−1 and p2s+1 := p2s + 3ℓ2s; note, then, that +σp2s−1(d) = XsYs · · · XtYtwin(1 − in/2) +and +σp2s(d) = YsXs+1 · · · XtYtwin(1 − in/2). +Let k2s−1, k2s ∈ {1, . . . , n} be the indices for which +σp2s−1(d) = wik2s−1 · · · win−1win(1 − in/2) +and +σp2s(d) = wik2s · · · win−1win(1 − in/2). +Using (20) and (22), we compute +v(Ξ(d)) = v(1(02)ℓ1(030)ℓ2 · · · (02)ℓ2t−1(030)ℓ2t0201) += 1 +β + +t +� +s=1 +� +1 +βp2s−1 v((02)ℓ2s−1) + +1 +βp2s v((030)ℓ2s) +� ++ +1 +βm−3 v(0201) += 1 +β + +t +� +s=1 +� +2 +βp2s−1+1 (1 − 1/β2ℓ2s−1) + +3 +2βp2s (1 − 1/β3ℓ2s) +� ++ +1 +βm−3 (2/β2 + 1/β4). +Moreover, (21) gives +v(d1 · · · dm−3) = 1 +β + +t +� +s=1 +� +1 +βp2s−1 v(Xs) + +1 +βp2s v(Ys) +� += 1 +β + +t +� +s=1 +� +1 +βp2s−1 v(Xs) + +1 +βp2s v((001)ℓ2s) +� += 1 +β + +t +� +s=1 +� +1 +βp2s−1 v(Xs) + +1 +2βp2s+1 (1 − 1/β3ℓ2s) +� +, +so equation (19) becomes +Kd = − 1 +β + +� +1≤k≤n−1 +ik=2 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +ik=0 +v(1wi1 · · · wik) ++ +t +� +s=1 +� +1 +βp2s−1+1 v(Xs) + +1 +2βp2s+2 (1 − 1/β3ℓ2s) +� ++ +1 +βm−1 . +26 + +Then +βv(Ξ(d)) + 5Kd =1 + +t +� +s=1 +� +2 +βp2s−1 (1 − 1/β2ℓ2s−1) + +3β +2βp2s (1 − 1/β3ℓ2s) +� ++ +1 +βm−3 (2/β + 1/β3) +− 5 +β + 5 +� +� +� +� +� +1≤k≤n−1 +ik=2 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +ik=0 +v(1wi1 · · · wik) +� +� +� +� ++ 5 +t +� +s=1 +� +1 +βp2s−1+1 v(Xs) + +1 +2βp2s+2 (1 − 1/β3ℓ2s) +� ++ +5 +βm−1 +=1 − 5 +β + +t +� +s=1 +1 +βp2s +� +3β/2 + 5/2β2� +(1 − 1/β3ℓ2s) + +1 +βm−3 (2/β + 5/β2 + 1/β3) ++ +t +� +s=1 +� +2 +βp2s−1 (1 − 1/β2ℓ2s−1) + +5 +βp2s−1+1 v(Xs) +� ++ 5 +� +� +� +� +� +1≤k≤n−1 +ik=2 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +ik=0 +v(1wi1 · · · wik) +� +� +� +� . +One easily verifies that both 3β/2 + 5/2β2 and 2/β + 5/β2 + 1/β3 equal c := 5 − β. We claim that it suffices +to show that +t +� +s=1 +� +2 +βp2s−1 (1 − 1/β2ℓ2s−1) + +5 +βp2s−1+1 v(Xs) +� +(25) ++ 5 +� +� +� +� +� +1≤k≤n−1 +ik=2 +v(1wi1 · · · wik) − +� +1≤k≤n−1 +ik=0 +v(1wi1 · · · wik) +� +� +� +� +≤ +t +� +s=1 +c +βp2s−1 (1 − 1/β2ℓ2s−1), +with equality if and only if d ≺ 1(w2w0)∞. Indeed, suppose the claim holds. Then the computation above +becomes +βv(Ξ(d)) + 5Kd ≤ 1 − 5 +β + c +t +� +s=1 +� +1 +βp2s−1 (1 − 1/β2ℓ2s−1) + +1 +βp2s (1 − 1/β3ℓ2s) +� ++ +c +βm−3 += 1 − 5 +β + c +t +� +s=1 +(1/βp2s−1 − 1/βp2s + 1/βp2s − 1/βp2s+1) + +c +βm−3 += 1 − 5 +β + c(1/β − 1/βm−3) + +c +βm−3 += 1 − 5 +β + c +β += 0 +with equality if and only if d ≺ 1(w2w0)∞. Rearranging, this inequality is equivalent to Kd/βv(Ξ(d)) ≤ +−1/5. From (18) and the assumption that n(d) = 1, this gives +fS(α) = 1 + Kd/βv(Ξ(d)) ≤ 4/5 +with equality if and only if d ≺ 1(w2w0)∞, as desired. +27 + +It remains to show the claim from (25). The constant c defined above may be rewritten as c = 2+5/(β2+1). +Subtracting �t +s=1(2/βp2s−1)(1 − 1/β2ℓ2s−1) from both sides, dividing by 5 and noting that ik ∈ {0, 2} only +when k2s−1 ≤ k < k2s, 1 ≤ s ≤ t, equation (25) becomes +t +� +s=1 +� +� +� +� +1 +βp2s−1+1 v(Xs) + +� +k2s−1≤k 0. We must show +ns,4j−3 +� +ℓ=1 +(v(d) − v(1X1Y1 · · · Xs−1Ys−1wns,1 +2 +· · · (w0w2)ns,4j−4wℓ +2)) > +1 +βps,4j−3 +1 +β2(β2 + 1)(1 − 1/β2ns,4j−3). +(29) +The left-hand side of the previous line equals +ns,4j−3 +� +ℓ=1 +� +1 +βps,4j−3+2ℓ v(wns,4j−3−ℓ +2 +(w2w0)ns,4j−2 · · · win(1 − in/2)) +� +. +29 + +Note that (w2w0)ns,4j−2 · · · win(1−in/2) ≻ (w0w2)∞: if not, then the former word begins with (w0w2)n′w0wi +for some n′ ≥ 0 and i ∈ {0, 1}. But then +wns,4j−3 +2 +(w2w0)ns,4j−2 · · · win(1 − in/2) = wns,4j−3 +2 +(w0w2)n′w0wi = wns,4j−3−1 +2 +(w0w2)n′+1wi, +contradicting the minimality of the sum of powers �4rs +ℓ=1 ns,ℓ. Thus the left-hand side of (29) is strictly +greater than +ns,4j−3 +� +ℓ=1 +� +1 +βps,4j−3+2ℓ v(wns,4j−3−ℓ +2 +) + +1 +βps,4j−2 v((w0w2)∞) +� += +1 +βps,4j−3 +ns,4j−3 +� +ℓ=1 +� +1 +β2ℓ+1 (1 − 1/β2ns,4j−3−2ℓ) + +1 +β2 + 1 +1 +β2ns,4j−3+1 +� += +1 +βps,4j−3 +� 1 +β2 (1 − 1/β2ns,4j−3) − +ns,4j−3 +β2ns,4j−3+1 + +1 +β2 + 1 +ns,4j−3 +β2ns,4j−3+1 +� += +1 +βps,4j−3 +� 1 +β2 (1 − 1/β2ns,4j−3) − +β2 +β2 + 1 +ns,4j−3 +β2ns,4j−3+1 +� +. +It suffices to show that the right-hand side of the previous line is greater than or equal to the right-hand +side of (29). Multiplying both quantities by βps,4j−3+2(β2 + 1), this is equivalent to showing +(β2 + 1)(1 − 1/β2ns,4j−3) − β3 ns,4j−3 +β2ns,4j−3 ≥ 1 − +1 +β2ns,4j−3 , +which simplifies to +1 − +1 +β2ns,4j−3 ≥ β ns,4j−3 +β2ns,4j−3 . +The left- and right-hand sides of the previous line increase and decrease, respectively, as functions of integers +ns,4j−3 > 0. Since the inequality holds for ns,4j−3 = 1, we conclude that (29) holds. Thus the summand on +the left-hand side of (28) is less than or equal to ns,4j−3v(d), with equality if and only if ns,4j−3 = 0. +Next, consider the summand on the right-hand side of (28). We shall show that this is greater than or +equal to ns,4j−1v(d) with equality if and only if ns,4j−1 = 0. Again if ns,4j−1 = 0, both the summand and +ns,4j−1v(d) equal zero, so assume ns,4j−1 > 0. The desired inequality is equivalent to +ns,4j−1(v(d) − v(1X1Y1 · · · Xs−1Ys−1wns,1 +2 +· · · (w2w0)ns,4j−2)) < +1 +βps,4j−1 +1 +β2 + 1(1 − 1/β2ns,4j−1). +(30) +The left-hand side of the previous line equals +ns,4j−1 +βps,4j−1 v(wns,4j−1 +0 +(w0w2)ns,4j · · · win(1 − in/2)) = ns,4j−1 +βps,4j v((w0w2)ns,4j · · · win(1 − in/2)). +For similar reasons as above, one finds that (w0w2)ns,4j · · · win(1 − in/2) ≺ (w2w0)∞. It follows that the +left-hand side of (30) is strictly less than +ns,4j−1 +βps,4j v((w2w0)∞) = ns,4j−1 +βps,4j +β +β2 + 1. +Multiplying both sides of (30) by βps,4j(β2 + 1) and recalling that ps,4j − ps,4j−1 = 2ns,4j−1, it thus suffices +to show that +βns,4j−1 ≤ β2ns,4j−1 − 1, +which clearly holds for each ns,4j−1 ≥ 1. This proves that the summand on the right-hand side of (28) is +greater than or equal to ns,4j−1v(d) with equality if and only if ns,4j−1 = 0. +Note that (17) may be rewritten as +n(d) = (1 + #{1 ≤ k ≤ n − 1 | ik ∈ {1, 2}} + 1) − (#{1 ≤ k ≤ n − 1 | 2 − ik ∈ {1, 2}} + 1) += 1 + #{1 ≤ k ≤ n − 1 | ik = 2} − #{1 ≤ k ≤ n − 1 | ik = 0}. +Since n(d) = 1 by assumption, we have +#{1 ≤ k ≤ n − 1 | ik = 2} = #{1 ≤ k ≤ n − 1 | ik = 0}. +30 + +Recalling that d = 1X1Y1 · · · XtYtwin(1 − in/2), (27) and the fact that each Ys consists solely of w1’s, we +find +#{1 ≤ k ≤ n − 1 | ik = 2} = +t +� +s=1 +rs +� +j=1 +(ns,4j−3 + ns,4j−2 + ns,4j) +and +#{1 ≤ k ≤ n − 1 | ik = 0} = +t +� +s=1 +rs +� +j=1 +(ns,4j−2 + ns,4j−1 + ns,4j), +so +t +� +s=1 +rs +� +j=1 +ns,4j−3 = +t +� +s=1 +rs +� +j=1 +ns,4j−1. +(31) +Using this and our prior observations regarding the left- and right-hand sides of (28), we have +t +� +s=1 +rs +� +j=1 +� ns,4j−3 +� +ℓ=1 +v(1X1Y1 · · · Xs−1Ys−1wns,1 +2 +· · · (w0w2)ns,4j−4wℓ +2) ++ +1 +βps,4j−3 +1 +β2(β2 + 1)(1 − 1/β2ns,4j−3) +� +≤v(d) +t +� +s=1 +rs +� +j=1 +ns,4j−3 +=v(d) +t +� +s=1 +rs +� +j=1 +ns,4j−1 +≤ +rs +� +j=1 +� +ns,4j−1v(1X1Y1 · · · Xs−1Ys−1wns,1 +2 +· · · (w2w0)ns,4j−2) + +1 +βps,4j−1 +1 +β2 + 1(1 − 1/β2ns,4j−1) +� +with equality throughout if and only if each ns,4j−1 = ns,4j−3 = 0. Thus the inequality in (28)—and hence +in (25)—holds. It remains to show that d ≺ 1(w2w0)∞ if and only if each ns,4j−1 = ns,4j−3 = 0. +Suppose that d ⪰ 1(w2w0)∞. Then d begins with 1(w2w0)n′w2wi for some n′ ≥ 0 and i ∈ {1, 2}. This +implies that either n1,1 or n1,5 is positive. For the converse, suppose that some ns,4j−3 or ns,4j−1 is positive. +By (31), we can choose some ns,4j−3 > 0 with (s, j) (lexicographically) minimal. Note that +σps,4j−3−1(d) = diwns,4j−3 +2 +(w2w0)ns,4j−2 · · · win(1 − in/2) +with di ∈ {0, 1}. Suppose di = 0 (the case that di = 1 is similar). Then j > 1, and +σps,4j−7(d) = wns,4j−7 +2 +(w2w0)ns,4j−6wns,4j−5 +0 +(w0w2)ns,4j−4wns,4j−3 +2 +(w2w0)ns,4j−2 · · · win(1 − in/2). +Since di = 0, we must have ns,4j−4 = 0. Moreover, ns,4j−5 > 0 contradicts the minimality of �4rs +ℓ=1 ns,ℓ, so +ns,4j−5 = 0. Since no three consecutive ns,ℓ’s can be zero (except possibly the first and final three ns,ℓ), it +follows that ns,4j−6 > 0. Thus +σps,4j−6−1(d) = di′(w2w0)ns,4j−6wns,4j−3 +2 +(w2w0)ns,4j−2 · · · win(1 − in/2) +(32) +for some di′ ∈ {0, 1}. Suppose di′ = 0. Then j > 2, and +σps,4j−11(d) =wns,4j−11 +2 +(w2w0)ns,4j−10wns,4j−9 +0 +(w0w2)ns,4j−8wns,4j−7 +2 +(w2w0)ns,4j−6wns,4j−3 +2 +(w2w0)ns,4j−2 +· · · win(1 − in/2). +Since di′ = 0, we have ns,4j−8 = n4j−7 = 0. If ns,4j−9 > 0, the fact that ns,4j−3 > 0 contradicts the +minimality of �4rs +ℓ=1 ns,ℓ. But ns,4j−9 = 0 is also a contradiction since this implies three consecutive ns,ℓ’s +are zero. Thus di′ = 1. Now suppose σps,4j−6−1(d) ≺ 1(w2w0)∞. From (32), we find that ns,4j−3 = 1 and +σps,4j−6−1(d) = 1(w2w0)ns,4j−6w2(w0w2)n′w0wi′′ · · · win(1 − in/2) +31 + +for some n′ ≥ 0 and i′′ ∈ {0, 1}. In any case, this contradicts the minimality of �4rs +ℓ=1 ns,ℓ. Thus (using the +fact that d ∈ M), +d ⪰ σps,4j−6−1(d) ⪰ 1(w2w0)∞, +and we conclude that d ≺ 1(w2w0)∞ if and only if each ns,4j−1 = ns,4j−3 = 0. +□ +Note that for each n ≥ 1, the word dn := 1(w2w0)n001 ≺ 1(w2w0)∞ satisfies Property M. Moreover, +v(dn) approaches v(1(w2w0)∞) = 2β/(β+2) from below, and thus 1/v(dn) approaches (β+2)/(2β) = 1/2+ +1/β from above. If d ∈ M satisfies d ≺ 1(w2w0)∞, then there is some n ≥ 1 for which d ≺ dn ≺ 1(w2w0)∞ +and 1/2 + 1/β < 1/v(dn) < 1/v(d). Since Idn ∩ Id = ∅ and Idn and Id contain 1/v(dn) and 1/v(d), +respectively, it follows that Id ⊂ (1/2 + 1/β, β]. Similarly reasoning shows that if d ≻ 1(w2w0)∞, then +Id ⊂ (1, 1/2 + 1/β), and in fact 1/2 + 1/β is a non-matching parameter. +With these observations and the previous lemmas, we are now ready to prove the main result of this +section: +Theorem 3.7. The frequency functions fS, fT : [1, β] → [0, 1] attain their maximums fS(α) = 4/5 and +fT (α) = 3/4 on the maximal interval [1/2 + 1/β, 1 + 1/β2]. +Proof. By (15), it suffices to show the statement for fS. Recall from Example 3.4 that fS equals 4/5 on +I1010 ∪ I1001 = (1 + 1/β4, 1 + 1/β2)\{1 + 1/β3}. Moreover, fS is decreasing on I10 = (1 + 1/β2, β] since +n(10) = 0. By continuity of fS, the statement is proven for α ∈ [1 + 1/β4, β]. +We now show that fS(α) ≤ 4/5 for α ∈ [1, 1+1/β4), with equality if α ≥ 1/2+1/β. Since fS is continuous +and is monotone on each matching interval Id, and since the set of matching parameters ∪d∈MId is dense, +it suffices to show the desired statements for the endpoints α± +d of matching intervals in [1, 1 + 1/β4). Notice +that each endpoint α+ +d , α− +d ∈ [1, 1 + 1/β4) is the limit (from above) of some sequence of endpoints of cascade +intervals. In particular, if d ∈ ψ(MU), then Id is itself a cascade interval and we take constant sequences. +Suppose d ∈ MU\ψ(MU). Since each lower endpoint α− +d equals the upper endpoint α+ +ψ(d) of Iψ(d) by +Proposition 2.23, we can again take the constant sequence. Now consider α+ +d . Let ε > 0, and choose some +matching parameter α′ ∈ Id′ satisfying α+ +d < α′ < α+ +d +ε. Since matching intervals are disjoint, Proposition +2.23 implies that the cascade interval Iψ(d′) lies strictly between α+ +d and α′, and thus its endpoints are within +a distance of ε of α+ +d . It follows α+ +d is the limit (from above) of a sequence of endpoints of cascade intervals. +Again by continuity of fS, it now suffices to show the desired statements for endpoints of cascade intervals. +These follow directly from Lemmas 3.5 and 3.6 and the observation above that if Id ⊂ (1/2 + 1/β, β], then +d ≺ 1(w2w0)∞. +Maximality of the interval [1/2 + 1/β, 1 + 1/β2] follows from the fact that fS is strictly decreasing on +(1 + 1/β2, β], density of matching parameters in [1, β] and Lemmas 3.5 and 3.6. +□ +Theorem 1.2 is now a collection of previous results: +Proof of Theorem 1.2. This is a direct consequence of Proposition 3.3, Theorem 3.7 and Equations (15) and +(18). +□ +4. Appendix: proofs of technical lemmas +We include here two technical results, which together with Lemma 2.12 prove Lemma 2.17. Recall that +∆(u) denotes the cylinder set of points x ∈ [0, 1] for which the β-expansion of x begins with u. +Lemma 4.1. Let d = d1 · · · dm ∈ MU and e := ϕ(d) = e1 · · · em. The β-expansions of 1/α− +d , 1/α+ +d and +1 − 1/α+ +d are given by +(bj(1/α− +d ))j≥1 = +� +(de2 · · · em−20)∞, +dm = 1 +(de1 · · · em−10)∞, +dm = 0 , +(bj(1/α+ +d ))j≥1 = +� +(d1 · · · dm−10)∞, +dm = 1 +(d1 · · · dm−20)∞, +dm = 0 +and +(bj(1 − 1/α+ +d ))j≥1 = +� +e∞, +dm = 1 +0(e2 · · · em)∞, +dm = 0 . +32 + +Proof. We consider only the β-expansion of 1/α− +d for dm = 1; the proofs of the other expansions are similar. +It suffices to show that 1/α− +d ∈ ∆(de2 · · · em−20) and B2m−2(1/α− +d ) = 1/α− +d . First, note that +v(de2 · · · em−20) = v(d) − (1/βm)v(e2 · · · em−2) += v(d) − (1/βm−1)v(e1 · · · em−2) += v(d) − (1/βm−1)(v(e) + 1/βm−1) += v(d) − (1/βm−1)(v(d) − 1 + 1/βm−1) += (1 − 1/βm−1)(v(d) + 1/βm−1). +Using this and Equation (8), 1/α− +d ∈ ∆(de2 · · · em−20) if and only if +(1 − 1/βm−1)(v(d) + 1/βm−1) ≤ 1/α− +d < (1 − 1/βm−1)v(d) + 1/βm−1. +Since dm = 1, the first inequality holds if and only if +(1 − 1/βm−1)(v(d) + 1/βm−1) ≤ βmv(d) + β +βm + β +, +or +(βm + β)(1 − 1/βm−1)(v(d) + 1/βm−1) ≤ βmv(d) + β. +Factoring βm from the first and multiplying it through the third term, the left-hand side is equal to +(1 + 1/βm−1)(1 − 1/βm−1)(βmv(d) + β) = (1 − 1/β2m−2)(βmv(d) + β), +which is less than βmv(d) + β. The second inequality is true if and only if +βmv(d) + β +βm + β +< (1 − 1/βm−1)v(d) + 1/βm−1. +Multiplying both sides by βm + β, this is equivalent to +βmv(d) + β < (βm − 1/βm−2)v(d) + β + 1/βm−2, +or (1/βm−2)v(d) < 1/βm−2. This holds since v(d) < v((10)∞) = 1. Thus 1/α− +d ∈ ∆(de2 · · · em−20). With +this and Equation (6), +B2m−2(1/α− +d ) = β2m−2(1/α− +d − v(de2 · · · em−20)) += β2m−2 +�βmv(d) + β +βm + β +− (1 − 1/βm−1)(v(d) + 1/βm−1) +� += β2m−2 +�βmv(d) + β − (βm − 1/βm−2)(v(d) + 1/βm−1) +βm + β +� += βmv(d) + β +βm + β += 1/α− +d . +□ +Lemma 4.2. Let d = d1 · · · dm ∈ MU and e := ϕ(d) = e1 · · · em. If dm = 1, then for each j > 0, +σj((de2 · · · em−20)∞) ⪯ (de2 · · · em−20)∞ +and +σj(e∞) ⪯ (d1 · · · dm−10)∞. +If dm = 0, then for each j > 0, +σj((de1 · · · em−10)∞) ⪯ (de1 · · · em−10)∞ +and +σj(0(e2 · · · em)∞) ⪯ (d1 · · · dm−20)∞. +33 + +Proof. We prove the statements for dm = 1; the other proofs are similar. Write +d = 1wi1wi2 · · · win(1 − in/2) +and +e = 0w2−i1w2−i2 · · · w2−in(in/2) +with each ik ∈ {0, 1, 2} and in = 0. Due to periodicity, it suffices to show the first inequality for 0 ≤ j < m−2. +Note that dm = 1 implies em−1 = 1. If j ≥ m, then +σj((de2 · · · em−20)∞) = (ej−m+2 · · · em−20de2 · · · ej−m+1)∞ ≺ ej−m+2 · · · em−1em ⪯ d ≺ (de2 · · · em−20)∞. +Now suppose 0 ≤ j < m. It suffices to show that +dj+1 · · · dme2 · · · em−20d1 · · · dj ⪯ de2 · · · em−20. +This trivially holds if j = 0, so assume j > 0. Since σj(d) ⪯ d, we have dj+1 · · · dm ⪯ d1 · · · dm−j. If this +inequality is strict, we are finished. Suppose equality holds. Then we wish to show +e2 · · · em−20d1 · · · dj ⪯ dm−j+1 · · · dme2 · · · em−20. +Since em−1 = 1, it suffices to show +e2 · · · em−1 ⪯ dm−j+1 · · · dme2 · · · em−j−1. +(33) +If j = m − 1, this is trivial, so suppose j < m − 1. By assumption, dj+1 · · · dm = d1 · · · dm−j, so dj+1 = +d1 = 1 = dm = dm−j. Now d2 = 0 implies j ̸= m − 2, and similarly dm−2 = 0 implies j ̸= m − 3. Hence +j < m − 3, and dj+1 = dm−j = 1 imply that dj+2 and dm−j+1 are the beginnings of some blocks wip and +wiℓ, respectively. (Similarly, ej+2 and em−j+1 are the beginnings of w2−ip and w2−iℓ, respectively.) Then +d1 · · · dm−j = dj+1 · · · dm may be written as +1wi1 · · · wiℓ−1 = 1wip · · · win−1001. +In particular, iℓ−1 = 1, and w2−iℓ−1 = w1 implies em−j = 1. +The desired inequality (33) may be written in terms of blocks: +w2−i1 · · · w2−in ⪯ wiℓ · · · win−1w1w2−i1 · · · w2−iℓ−2. +Suppose for the sake of contradiction that this inequality does not hold, and let 1 ≤ k ≤ n be minimal such +that w2−ik differs from the kth block on the right-hand side. Then +w2−ik ≻ +� +� +� +� +� +wiℓ+k−1, +k < n − ℓ + 1 +w1, +k = n − ℓ + 1 +w2−ik−(n−ℓ)−1, +k > n − ℓ + 1 +, +and we consider these three cases separately: +(i) If k < n − ℓ + 1, then +(2 − i1, . . . , 2 − ik−1) = (iℓ, . . . , iℓ+k−2) +and 2 − ik > iℓ−k−1 imply +(2 − iℓ, . . . , 2 − iℓ+k−2) = (i1, . . . , ik−1) +and 2 − iℓ−k−1 > ik. This gives +1w2−iℓ · · · w2−iℓ+k−1 ≻ 1wi1 · · · wik. +Recall that em−j+1 is the beginning of the block w2−iℓ, so the previous line together with em−j = 1 +imply σm−j−1(e) ≻ d, a contradiction. +(ii) If k = n − ℓ + 1, then +(2 − i1, . . . , 2 − in−ℓ) = (iℓ, . . . , in−1) +and 2 − in−ℓ+1 > 1 imply +(2 − iℓ, . . . , 2 − in−1) = (i1, . . . , in−ℓ) +and in−ℓ+1 = 0. Since 2 − in = 2, this implies +1w2−iℓ · · · w2−in ≻ 1wi1 · · · win−ℓ+1. +34 + +As in case (i), this gives the contradiction that σm−j−1(e) ≻ d. +(iii) If k > n − ℓ + 1, then +(2 − i1, . . . , 2 − in−ℓ) = (iℓ, . . . , in−1) +and 2 − in−ℓ+1 = 1 implies +(2 − iℓ, . . . , 2 − in−1) = (i1 . . . , in−ℓ) +and in−ℓ+1 = 1. Again since 2 − in = 2, +1w2−iℓ · · · w2−in ≻ 1wi1 · · · win−ℓ+1, +and the contradiction of cases (i) and (ii) arises. +This proves for each j > 0 that +σj((de2 · · · em−20)∞) ⪯ (de2 · · · em−20)∞. +It remains to show that +σj(e∞) ⪯ (d1 · · · dm−10)∞, +or, equivalently, +ej+1 · · · eme1 · · · ej ⪯ d1 · · · dm−10 +for 0 ≤ j < m. Suppose for the sake of contradiction that this inequality does not hold. If there is some +k ≤ m − j for which +ej+1 · · · ej+k ≻ d1 · · · dk, +then σj(e) ≻ d, a contradiction. Thus there is some minimal 1 ≤ k ≤ j for which +ej+1 · · · eme1 · · · ek ≻ d1 · · · dm−j+k. +The previous line may be written in block form +1w2−iℓ · · · w2−in−1w2w0w2−i1 · · · w2−ip ≻ 1wi1 · · · wiq +for some ℓ, p, q ∈ {1, . . . n}. In particular, +(0, 2 − i1, . . . , 2 − ip−1) = (iq−p, iq−p+1, . . . , iq−1) +and 2 − ip > iq imply +(2 − iq−p, 2 − iq−p+1, . . . , 2 − iq−1) = (2, i1, . . . , ip−1) +and 2 − iq > ip. Since w2−iq−p = 01, there is some s ≥ 0 such that +σs(e) = 1w2−iq−p+1 · · · w2−iq−1w2−iq · · · w2−in0 ≻ 1wi1 · · · wip−1wip · · · win1 = d, +contrary to the assumption that d ∈ M. +□ +References +[1] Banerjee, S., Karthik, M. S., Yuan, G., and Yorke, J. Bifurcations in one-dimensional piecewise smooth maps—theory +and applications in switching circuits. IEEE Trans. Circuits Systems I Fund. Theory Appl. 47, 3 (2000), 389–394. +[2] Bonanno, C., Carminati, C., Isola, S., and Tiozzo, G. Dynamics of continued fractions and kneading sequences of +unimodal maps. Discrete Contin. Dyn. Syst. 33, 4 (2013), 1313–1332. +[3] Botella-Soler, V., Oteo, J. A., and Ros, J. Dynamics of a map with a power-law tail. J. Phys. A 42, 38 (2009), +385101, 22. +[4] Botella-Soler, V., Oteo, J. A., Ros, J., and Glendinning, P. Lyapunov exponent and topological entropy plateaus +in piecewise linear maps. J. Phys. A 46, 12 (2013), 125101, 26. +[5] Bruin, H., Carminati, C., and Kalle, C. Matching for generalised β-transformations. Indag. Math. (N.S.) 28, 1 (2017), +55–73. +[6] Bruin, H., Carminati, C., Marmi, S., and Profeti, A. Matching in a family of piecewise affine maps. Nonlinearity 32, +1 (2019), 172–208. +[7] Carminati, C., Marmi, S., Profeti, A., and Tiozzo, G. The entropy of α-continued fractions: +numerical results. +Nonlinearity 23, 10 (2010), 2429–2456. +[8] Carminati, C., and Tiozzo, G. A canonical thickening of Q and the entropy of α-continued fraction transformations. +Ergodic Theory Dynam. Systems 32, 4 (2012), 1249–1269. +[9] Carminati, C., and Tiozzo, G. Tuning and plateaux for the entropy of α-continued fractions. Nonlinearity 26, 4 (2013), +1049–1070. +[10] Chen, Y., and Kraaikamp, C. Matching of orbits of certain N-expansions with a finite set of digits. arXiv:2209.08882v1, +2022. +35 + +[11] Cholewa, �L., and Oprocha, P. Renormalization in lorenz maps – completely invariant sets and periodic orbits. +arXiv:2104.00110v2, 2021. +[12] Cosper, D., and Misiurewicz, M. Entropy locking. Fund. Math. 241, 1 (2018), 83–96. +[13] Dajani, K., and Kalle, C. Invariant measures, matching and the frequency of 0 for signed binary expansions. Publ. Res. +Inst. Math. Sci. 56, 4 (2020), 701–742. +[14] Dajani, K., and Kalle, C. A first course in ergodic theory. CRC Press, Boca Raton, FL, 2021. +[15] Dajani, K., Kalle, C., and Maggioni, M. Matching for random systems with an application to minimal weight expan- +sions. Nonlinearity 34, 6 (2021), 3676–3708. +[16] Dajani, K., Kraaikamp, C., and Steiner, W. Metrical theory for α-Rosen fractions. J. Eur. Math. Soc. (JEMS) 11, 6 +(2009), 1259–1283. +[17] Kopf, C. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput. 39, 2 (1990), +123–144. +[18] Kraaikamp, C., Schmidt, T., and Steiner, W. Natural extensions and entropy of α-continued fractions. Nonlinearity +25, 8 (2012), 2207–2243. +[19] Lasota, A., and Yorke, J. A. On the existence of invariant measures for piecewise monotonic transformations. Trans. +Amer. Math. Soc. 186 (1973), 481–488. +[20] Nakada, H., and Natsui, R. The non-monotonicity of the entropy of α-continued fraction transformations. Nonlinearity +21, 6 (2008), 1207–1225. +[21] Parry, W. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. +[22] R´enyi, A. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. +36 + diff --git a/5NFAT4oBgHgl3EQfmh0R/content/tmp_files/load_file.txt b/5NFAT4oBgHgl3EQfmh0R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23326a90c8d187132e5a5108e5809ada83865345 --- /dev/null +++ b/5NFAT4oBgHgl3EQfmh0R/content/tmp_files/load_file.txt @@ -0,0 +1,1405 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf,len=1404 +page_content='ERGODIC PROPERTIES OF A PARAMETERISED FAMILY OF SYMMETRIC GOLDEN MAPS: THE MATCHING PHENOMENON REVISITED KARMA DAJANI AND SLADE SANDERSON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We study a one-parameter family of interval maps {Tα}α∈[1,β], with β the golden mean, defined on [−1, 1] by Tα(x) = β1+|t|x − tβα where t ∈ {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each Tα, α > 1, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters α, the corresponding density is a step function with finitely many jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We give an explicit description of the maximal intervals of parameters on which the density has at most the same number of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A main tool in our analysis is the phenomenon of matching, where the orbits of the left and right limits of discontinuity points meet after a finite number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Each Tα generates signed expansions of numbers in base 1/β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' via Birkhoff’s ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies of digits in generic Tα-expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, the frequency of 0 is shown to vary continuously as a function of α and to attain its maximum 3/4 on the maximal interval [1/2 + 1/β, 1 + 1/β2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Introduction Dynamical systems given by piecewise monotone maps T : I → I of an interval have a rich history: besides having applications in various fields—including population ecology ([3]) and controlled switching circuits ([1])—these systems are often used to produce expansions of numbers from the underlying interval I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Examples include decimal, n-ary, continued fraction, (generalised) L¨uroth and β-expansions, though this list is far from exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A common theme in the study of these expansions is the investigation of asymptotic relative frequencies of digits occurring in typical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lebesgue–almost all) expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' To this end, the standard procedure is the construction of an ergodic, T-invariant measure µ equivalent to Lebesgue measure λ and a calculation of the µ-measure of the subinterval of I corresponding to the digit(s) in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Birkhoff’s ergodic theorem asserts that the measure of this subinterval equals the desired asymptotic frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In [13], invariant measures and frequencies of digits are studied for a family of symmetric doubling maps {Dη}η∈[1,2] defined on [−1, 1] by Dη(x) = 2x − d(x)η with d(x) ∈ {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These maps produce signed binary expansions of numbers x ∈ [−1, 1] of the form x = η � n≥1 dn/2n with each dn ∈ {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It is shown that each Dη, η > 1, admits an ergodic, invariant measure equivalent to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The authors use a curious property called matching—defined in the sequel—to prove that there is a countable collection of disjoint, open subintervals of [1, 2] whose union has full measure, and such that on each such subinterval, the densities of the corresponding invariant measures are step functions with at most the same, finite number of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These explicitly constructed measures are then used to study the asymptotic frequency of the digit 0 in generic expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This frequency is shown to be continuous as a function of η and attains a maximal value of 2/3 on the maximal interval [6/5, 3/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, the frequency function is either constant, strictly increasing or strictly decreasing on each of the aforementioned subintervals of [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The present article continues these themes of inquiry with a parameterised family of skewed symmetric golden maps {Tα}α∈[1,β], with β = ( √ 5 + 1)/2 the golden mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' the positive real solution to β2 = β + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Department of Mathematics, Utrecht University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Box 80010, 3508TA Utrecht, The Netherlands E-mail addresses: k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='dajani1@uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='nl and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='sanderson@uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Date: January 20, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 37E05 (Primary) 28D05, 37A05 (Secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' invariant measure, ergodic theory, matching, interval map, number expansions, digit frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='08623v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='DS] 20 Jan 2023 Each Tα : [−1, 1] → [−1, 1] is defined by Tα(x) := � � � � � β2x + βα, x ∈ [−1, −1/β) βx, x ∈ [−1/β, 1/β] β2x − βα, x ∈ (1/β, 1] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Setting J−1 := [−1, −1/β), J0 := [−1/β, 1/β] and J1 := (1/β, 1], the map Tα may be written more succinctly as Tα(x) = β1+|t(x)|x − t(x)βα, (1) where t(x) ∈ {−1, 0, 1} is the unique index for which x ∈ Jt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For j ≥ 1, set tα,j(x) := t(T j−1 α (x));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' the sequence of digits (tα,j(x))j≥1 ∈ {−1, 0, 1}N records indices of the subsequent subintervals J−1, J0 or J1 entered by the forward orbit of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' With this notation, equation (1) gives for each j ≥ 1 T j α(x) = β1+|tα,j(x)|T j−1 α (x) − tα,j(x)βα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Solving this for T j−1 α (x), induction shows that for any n ≥ 1, x = α n � j=1 tα,j(x) βj−1+�j k=1 |tα,k(x)| + T n α (x) βn+�n k=1 |tα,k(x)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Taking the limit n → ∞ and recalling that |T n α (x)| ≤ 1 gives x = α � j≥1 tα,j(x) βj−1+�j k=1 |tα,k(x)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that for fixed α, this process determines a unique expansion for each x ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We refer to both this expansion and the corresponding sequence of digits (tα,j(x))j≥1 as the Tα-expansion of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Phenomena analogous to those observed in [13] are found to occur for the skewed symmetric binary maps Tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, we prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each α ∈ (1, β], the map Tα has a unique—hence ergodic—absolutely continuous in- variant probability measure µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, µα is equivalent to Lebesgue measure λ, and there is a countable collection {Id}d∈M of disjoint open subintervals of [1, β] of full Lebesgue measure, such that for fixed d ∈ M the density of each µα with α ∈ Id is a step function with at most the same, finite number of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Via Birkhoff’s ergodic theorem, these measures are employed to show the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The asymptotic relative frequency of the digit 0 in Lebesgue-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Tα-expansion depends continuously on α ∈ [1, β] and attains a maximum value of 3/4 on the (maximal) interval [1/2+1/β, 1+1/β2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Furthermore, the frequency function is either constant, strictly increasing or strictly decreasing on each Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' As in [13], the main tool used to construct the Tα-invariant measures is a property called matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' An interval map T : I → I is said to have matching if for each critical point c ∈ I, the orbits of the left and right limits y± := limx→c± T(x) agree after some finite number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 That is, for each critical point c ∈ I there are integers M, N ≥ 0 for which T M(y−) = T N(y+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Matching has gained considerable attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Intricacies of the metric entropy function of Nakada’s α-continued fraction maps have been studied using matching in [20], [7], [8], [18], [2] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, matching is used in [18] to determine the natural extension for each α-continued fraction transformation, and it is shown that the set of α ∈ [0, 1] for which matching does not occur has zero Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The Lebesgue measure of this set of non-matching parameters—in addition to the fact that its Hausdorff dimension is 1—is also shown in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Matching is used in [16] to determine invariant measures for the related family of α-Rosen continued fraction transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A parameterised family of linear maps with one increasing and one decreasing branch are considered in [4], and matching is used to show that in some parameter regions, the Lyapunov exponent and topological entropy are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A geometric explanation of matching for a similar family of maps is given in [12], and further implications of matching for these maps—including smoothness of entropy on an open dense subset of parameters—is considered in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 1Some authors require that the one-sided derivatives also agree at these times, in which case the map may be said to have strong matching ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This extra condition is not needed for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 2 The notion of matching is extended to random dynamical systems in [15] and is used to study the asymptotic frequency of the digit 0 in typical signed binary expansions arising from a family of random interval maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Matching has also been investigated for generalised β-transformations, a certain class of continued fraction expansions with finite digit sets, and Lorenz maps (see [5], [10] and [11], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The present paper exploits the phenomenon of matching in a fashion similar to that of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' There the authors use results of [17], which gives formulas for densities of the absolutely continuous invariant measures of piecewise linear expanding interval maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These densities are—in general—infinite sums of (finite) step functions which are determined by the orbits of the left and right limits at critical points of the underlying interval map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' However, when matching occurs the infinite sum becomes finite, and the density itself is a finite step function depending only on these orbits before matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In [13], it is shown that matching occurs for the symmetric doubling map Dη on a set of parameters η in [1, 2] of full Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For these matching parameters, the orbits of the left and right limits at the critical points before matching are studied in detail, and this information is used to provide an explicit formula for the density of the (unique) absolutely continuous invariant probability measure for each Dη with matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The parameter space [1, 2] is divided into a countable union of (maximal) open intervals—called matching intervals—where each Dη has matching, and a Lebesgue-null set of non-matching parameters with Hausdorff dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' On each matching interval, matching occurs after the same number of steps, and for each left/right limit at a critical point, the digits of the corresponding signed binary expansions agree before matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' While the results of the present paper imply that the same direct approach of understanding matching for the skewed symmetric golden maps Tα can be applied to construct the invariant measures asserted in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, we find that the unequal slopes of the different branches present difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' To circumvent these, we instead study matching for a family of symmetric golden maps {Sα}α∈[1,β] of constant slope for which the skewed symmetric golden maps {Tα}α∈[1,β] are jump transformations, and it is subsequently shown that the parameters α for which the maps Tα and Sα have matching coincide (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Equipped with this result, one could then use the formulas from [17] to determine invariant densities for the Tα with matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' however, we proceed in the simpler setting of the symmetric maps Sα—determining invariant densities and the frequencies of digits for these—and finally use the fact that Tα is the jump transformation of Sα to determine invariant measures and frequencies of digits for the original skewed symmetric golden maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In §2 the symmetric golden maps {Sα}α∈[1,β] are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These are shown in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 to have matching for Lebesgue–a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' α ∈ [1, β], and we also prove here that the matching parameters of both families {Sα}α∈[1,β] and {Tα}α∈[1,β] coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3 are devoted to understanding the finer structure of the set of matching parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The former provides a classification of all matching intervals and of the orbits of all left and right limits at critical points before matching occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In the latter, it is shown that all (but two) of the matching intervals generate in a natural fashion a whole ‘cascade’ of countably many matching intervals with adjacent endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In §3 we use the results of the preceding section to prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, explicit formulas for densities of the unique, absolutely continuous invariant measures of the symmetric golden maps Sα are provided in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, and the invariant measures of the skewed maps Tα are expressed in terms of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These measures are used in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 to determine expressions for the asymptotic frequencies of the digit 0 in typical Sα- and Tα-expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The maximal frequencies of the digit 0 as functions of α are considered in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='. Proofs of some technical results are provided in an appendix (§4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This work is part of project number 613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='135 of the research programme Mathe- matics Clusters which is financed by the Dutch Research Council (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Symmetric golden maps Sα As mentioned in §1, we determine invariant measures and the frequencies of digits for a family of symmetric golden maps {Sα}α∈[1,β] for which the {Tα}α∈[1,β] are jump transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These invariant measures and frequencies are then used to determine the invariant measures and frequencies of digits for the original Tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The maps Sα are defined as follows: for α ∈ [1, β], let Sα : [−1, 1] → [−1, 1] be given by Sα(x) := βx − t(x)α, 3 1 −1/β 0 1/β 1 1 0 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The maps Tα (blue) and Sα (red) with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that Tα = Sα on the middle interval J0 = [−1/β, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' with t(x) ∈ {−1, 0, 1} as in §1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that Sα(x) ∈ J0 for each x ∈ J−1 ∪ J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Using this, one readily verifies that Tα(x) = � Sα(x), x ∈ J0 S2 α(x), x ∈ J−1 ∪ J1 , (2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Tα is the jump transformation for Sα with respect to the sweep-out set J0 = [−1/β, 1/β] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 of [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each j ≥ 1, let sα,j(x) := t(Sj−1 α (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' With induction one finds that for each k ≥ 0, Sk α(x) = βk � �x − α k � j=1 sα,j(x)/βj � � (3) (with the summation for k = 0 understood to be 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since |Sk α| ≤ 1, dividing both sides by βk and taking the limit as k approaches infinity gives x = α � j≥1 sα,j(x)/βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (4) Following our convention from §1, we refer to both the right-hand side of Equation (4) and the corresponding sequence (sα,j(x))j≥1 of digits in {0, ±1}N as the Sα-expansion of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Again this process determines—for fixed α—a unique expansion for each x ∈ [−1, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' moreover, if x, y ∈ [−1, 1] have the same Sα-expansion, then Equation (3) can be used to show that x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Also note that not every sequence in {0, ±1}N is an Sα-expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' in particular, a 1 or −1 is necessarily followed by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' As the orbits of 1 and 1−α will be studied in detail below, we fix special notation for their Sα-expansions: let dα,j := sα,j(1) and eα,j := sα,j(1 − α) for each α ∈ [1, β] and j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' When α is understood, it is suppressed from the notation, and we simply write dj := dα,j and ej := eα,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Matching almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In this section, we show that the maps Sα (and Tα) have matching on a set of full Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 The map Sα has two critical points ±1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Due to symmetry, it suffices to consider the matching criteria only for the positive critical point 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that limx→1/β− Sα(x) = 1 and limx→1/β+ Sα(x) = 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Hence Sα has matching if and only if there are integers M, N ≥ 1 for which SM α (1) = SN α (1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We begin by investigating matching in a number of specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' First, note that 1 ∈ J1 and 1 − α ∈ J0 for all α ∈ [1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 2The general approach to proving this result largely follows that of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 of [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' however, we shall see that the dynamics of the symmetric golden maps Sα are—in a sense—more delicate than those of the previously studied symmetric binary maps (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 below with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 of [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 4 (i) If α ∈ (1 + 1/β2, β], then Sα(1) = β − α ∈ [0, 1/β3) ⊂ J0, Sα(1 − α) = β − βα ∈ [−1, −1/β) ⊂ J−1, S2 α(1) = β2 − βα ∈ J0 and S2 α(1 − α) = β2 − β2α + α = β2 − βα ∈ J0 shows that Sα has matching with M = N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (ii) If α = 1 + 1/β2, then Sα(1) = β − α = 1/β3 ∈ J0, Sα(1 − α) = β − βα = −1/β ∈ J0, S2 α(1) = 1/β2 ∈ J0, S2 α(1 − α) = −1 ∈ J−1, S3 α(1) = 1/β ∈ J0, S3 α(1 − α) = −1/β3 ∈ J0, S4 α(1) = 1 ∈ J1 and S4 α(1 − α) = −1/β2 = 1 − α ∈ J0, so Sα has a Markov partition, namely � [−1/β3/1/β3], ±(1/β3, 1/β2], ±(1/β2, 1/β], ±(1/β, 1] � , and no matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (iii) If α ∈ (1 + 1/β3, 1 + 1/β2), Sα(1) = β − α ∈ (1/β3, 1/β2) ⊂ J0, Sα(1 − α) = β − βα ∈ (−1/β, −1/β2) ⊂ J0, S2 α(1) = β2 − βα ∈ (1/β2, 1/β) ⊂ J0, S2 α(1 − α) = β2 − β2α ∈ (−1, −1/β) ⊂ J−1, S3 α(1) = β3 − β2α ∈ (1/β, 1) ⊂ J1, S3 α(1 − α) = β3 − (β3 − 1)α ∈ (−1/β3, 1/β3) ⊂ J0, S4 α(1) = β4 − (β3 + 1)α ∈ J0 and S4 α(1 − α) = β4 − (β4 − β)α ∈ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since β4 − β3 = β2 = β + 1, we find that S4(1) = S4(1 − α), so Sα has matching with M = N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (iv) If α = 1 + 1/β3, Sα(1) = β − α = 1/β2 ∈ J0, Sα(1 − α) = β − βα = −1/β2 ∈ J0, S2 α(1) = 1/β ∈ J0, S2 α(1 − α) = −1/β ∈ J0, S3 α(1) = 1 ∈ J1, S3 α(1 − α) = −1 ∈ J−1 and S4 α(1 − α) = −1/β2 ∈ J0, so Sα has a Markov partition and no matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (v) If α ∈ (1, 1 + 1/β3), then Sα(1) = β − α ∈ (1/β2, 1/β) ⊂ J0, Sα(1 − α) = β − βα ∈ (−1/β2, 0) ⊂ J0, S2 α(1) = β2 − βα ∈ (1/β, 1) ⊂ J1, S2 α(1 − α) = β2 − β2α ∈ (−1/β, 0) ⊂ J0, S3 α(1) = β3 − (β2 + 1)α ∈ (−1/β3, 1/β) ⊂ J0 and S3 α(1 − α) = β3 − β3α ∈ (−1, 0) ⊂ J−1 ∪ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This case will be considered more closely in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (vi) If α = 1, then Sα(1) = 1/β ∈ J0, S2 α(1) = 1 ∈ J1 and Sα(1 − α) = 0 = 1 − α ∈ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus there is a Markov partition and no matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that in the cases above in which there is matching—namely (i) and (iii)—we have M = N (a property called neutral matching in [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We shall see below that this is always the case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Sα has matching if and only if there is some m ≥ 1 for which Sm α (1) = Sm α (1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For this we need the following proposition—key to a number of arguments throughout—which states that the difference between subsequent points in the orbits of 1 and 1 − α can take on at most four values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that (dj)j≥1 and (ej)j≥1 denote the Sα-expansions of 1 and 1 − α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For every α ∈ [1, β] and j ≥ 0, Sj α(1) − Sj α(1 − α) ∈ {0, α/β, α, βα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For α /∈ (1, 1 + 1/β3), the statement is verified with the cases above, so assume α ∈ (1, 1 + 1/β3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We use induction on j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The result clearly holds for j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' assume for some j = k − 1 ≥ 0 that Sk−1 α (1) − Sk−1 α (1 − α) = y 5 for some y ∈ {0, α/β, α, βα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If y = 0, then also Sj α(1) − Sj α(1 − α) = 0 for all j ≥ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Suppose y ̸= 0, and note that Sk α(1) − Sk α(1 − α) = (βSk−1 α (1) − dkα) − (βSk−1 α (1 − α) − ekα) = βy − (dk − ek)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We determine the difference above for each y ∈ {α/β, α, βα}: (i) y = α/β: Since 1/β < y < 2/β, we have (dk, ek) = (1, 0), (0, −1) or (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In the first two cases Sk α(1) − Sk α(1 − α) = 0, and in the third Sk α(1) − Sk α(1 − α) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (ii) y = α: Since 1/β < y < 1 + 1/β3 = 2/β, we again have (dk, ek) = (1, 0), (0, −1) or (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In the first two cases Sk α(1) − Sk α(1 − α) = βα − α = α/β, and in the third Sk α(1) − Sk α(1 − α) = βα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (iii) y = βα: Since y > 2/β, we must have (dk, ek) = (1, −1), and hence Sk α(1) − Sk α(1 − α) = β2α − 2α = α/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The previous proposition can be used to give an equivalent definition of matching: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The map Sα has matching if and only if there is some m ≥ 1 for which Sm α (1) = Sm α (1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' One direction is immediate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' for the other, suppose there are distinct M, N ≥ 1 for which SM α (1) = SN α (1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume for the sake of contradiction that Sj α(1) ̸= Sj α(1 − α) for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, Sj α(1) − Sj α(1 − α) ≥ α/β ≥ 1/β, and hence Sj α(1 − α) ≤ Sj(1) − 1/β ≤ 1 − 1/β = 1/β2 for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If Sj α(1 − α) ∈ (0, 1/β2], then there is some k ≥ 0 for which Sj+k α (1 − α) = βkSj α(1 − α) > 1/β2, contradicting the above, and thus Sj α(1−α) ≤ 0 for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A similar argument implies Sj α(1) ≥ 0 for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' But SM α (1) = SN α (1 − α), so this common value must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since 0 is fixed by Sα, we have the contradiction that Sm α (1) = 0 = Sm α (1 − α) with m = max{M, N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ We can now define a canonical index to describe when matching occurs: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The matching index of Sα is m(α) := inf{m ≥ 1 | Sm α (1) = Sm α (1 − α)} ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The cases above together with the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 reveal a strong interdependence between the orbits of 1 and 1 − α, which is summarised in the graph of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, note that if matching occurs with matching index m := m(α), then Sm−1 α (1)−Sm−1 α (1−α) = α/β and (dm, em) ∈ {(1, 0), (0, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since Sα-expansions cannot contain consecutive non-zero digits, this implies Sm−2 α (1) − Sm−2 α (1 − α) = α and (dm−1, em−1) ∈ {(1, 0), (0, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For m > 2, this further implies Sm−3 α (1) − Sm−3 α (1 − α) = α/β and (dm−2, em−2) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus if Sα has matching with index m > 2, then the final three digits of the Sα-expansions of 1 and 1 − α before matching are given by �dm−2dm−1dm em−2em−1em � ∈ ��010 001 � , �001 010 �� , (5) where w := −w for w ∈ {0, ±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Conversely, if for some m > 2, three consecutive digits of the Sα-expansions of 1 and 1 − α are given by (5), then the proof implies that Sα has matching with index m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A number of characterisations of matching for Sα can be derived from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For these we fix some notation: for each x ∈ [−1, 1] and α ̸= 1, let ℓα(x) := inf j≥0{Sj α(|x|) ≤ 0} − 1, 6 0 α/β α βα � 0 1 � � 0 0 � � 1 0 � � 0 0 � � 0 0 � � 1 0 � � 0 0 � � 0 1 � � 0 0 � � 0 0 � � 1 0 � � 0 0 � � 1 0 � � 0 1 � � 0 1 � � 1 0 � � 0 1 � � 0 0 � � 0 0 � � 1 1 � � 1 1 � � 0 0 � � dj dj � � dj+1 dj+1 � Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A graphical representation of the interdependence of the orbits of 1 and 1 − α for α ∈ [1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Vertices represent the differences Sj−1 α (1) − Sj−1 α (1 − α) for j ≥ 1, and the beginnings and ends of edges are marked � dj ej � and � dj+1 ej+1 � , respectively, where w := −w for w ∈ {0, ±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Cyan edges are taken if and only if Sα has matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' and set ℓα := min{ℓα(1), ℓα(1 − α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For α ̸= 1, Sα has matching if and only if ℓα < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, if ℓα < ∞, then m(α) ∈ {ℓα + 1, ℓα + 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let ℓ := ℓα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' That matching implies ℓ < ∞ is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now suppose ℓ < ∞, and assume without loss of generality that ℓ = ℓα(1 − α) and thus Sℓ+1 α (1 − α) ≥ 0 (the other case is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The definitions of ℓ and m(α) give ℓ + 1 ≤ m(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, Sℓ+1 α (1 − α) ≥ 0 and α > 1 imply Sℓ+1 α (1) − Sℓ+1 α (1 − α) ∈ {0, α/β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The result holds if the difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If the difference is α/β, we must have (dℓ+2, eℓ+2) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From Figure 2, this implies Sℓ+2 α (1) − Sℓ+2 α (1 − α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For α ̸= 1, Sα has matching if and only if there exists some j ≥ 1 such that Sj α(1) ∈ (1/β, α/β] or Sj α(1 − α) ∈ [−α/β, −1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, ℓα(1) and ℓα(1 − α), respectively, are the infimums over all j for which the above inclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3 and the facts that S−1 α ([−1, 0]) ∩ (0, 1] = (1/β, α/β] and S−1 α ([0, 1]) ∩ [−1, 0) = [−α/β, 1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ Due to symmetry, the above corollary states that Sα has matching if and only if the orbit of either 1 or of α−1 enters the region (1/β, α/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We shall see that this occurs for Lebesgue–a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' α by relating the beginnings of these orbits to the beginnings of certain orbits of the (ergodic) β-transformation B : [0, 1] → [0, 1] defined by B(x) = βx (mod 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Set b(x) := � 0, x < 1/β 1, x ≥ 1/β , 7 and for each j ≥ 1, let bj(x) := b(Bj−1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We call the sequence (bj(x))j≥1 the β-expansion (also referred to as the greedy-expansion) of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Via induction, one finds that for each k ≥ 0, Bk α(x) = βk � �x − k � j=1 bj(x)/βj � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (6) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let x ∈ {1, α − 1}, α ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then (i) Sj α(x) = αBj(x/α) for each 0 ≤ j ≤ ℓα(x), (ii) sα,j(x) = bj(x/α) for each 1 ≤ j ≤ ℓα(x) and (iii) ℓα(x) is the infimum over all j for which Bj(x/α) ∈ (1/βα, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Claim (iii) will follow from claim (i), Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 and the fact that ℓα(x) = ℓα(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We prove claim (i) via induction on j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Certainly Sj α(x) = αBj(x/α) for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now suppose this equality holds for some j = k − 1 with 0 ≤ k − 1 < ℓα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4, Sk−1 α (x) ∈ [0, 1]\\(1/β, α/β], and we find Sk α(x) = � βSk−1 α (x), Sk−1 α (x) ∈ [0, 1/β] βSk−1 α (x) − α, Sk−1 α (x) ∈ (α/β, 1] = � βαBk−1(x/α), Bk−1 α (x/α) ∈ [0, 1/βα] βαBk−1(x/α) − α, Bk−1 α (x/α) ∈ (1/β, 1/α] = αBk(x/α), so the first claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Furthermore, the equality in (i) gives for each 1 ≤ j ≤ ℓα(x) that Sj−1 α (x) ∈ [0, 1/β] if and only if Bj−1(x/α) ∈ [1, 1/βα] and Sj−1 α (x) ∈ (α/β, 1] if and only if Bj−1(x/α) ∈ (1/β, 1/α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus sα,j(x) = bj(x/α) for such j, proving claim (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 and symmetry of Sα give yet another characterisation of matching in terms of the map B: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For α ̸= 1, Sα has matching if and only if there exists some j ≥ 0 such that Bj(1/α) ∈ (1/βα, 1/β] or Bj(1 − 1/α) ∈ (1/βα, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, ℓα(1) and ℓα(1 − α), respectively, are the infimums over all j for which the above inclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The previous results together with ergodicity of B can now be used to prove that Sα has matching for a set of parameters α of full Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The proof is nearly identical to that of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3 of [13] but is included here for the ease of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The map Sα has matching for Lebesgue–a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' α ∈ [1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let α ∈ (1, β] and k ∈ N with k > β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By ergodicity of B with respect to Lebesgue measure (§4 of [22]), for Lebesgue–a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' x ∈ [0, 1] there exists some j ≥ 1 such that Bj(x) ∈ (1/β − 1/k, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that 1/βα < 1/β − 1/k if and only if α > k/(k − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus for Lebesgue–a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' α ∈ (k/(k − β), β], there exists some j ≥ 1 such that Bj(1/α) ∈ (1/β − 1/k, 1/β] ⊂ (1/βα, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6, Sα has matching for Lebesgue–a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' α ∈ (k/(k − β), β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let Ak denote the set of α ∈ (k/(k − β), β] for which Sα does not have matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then ∪k>β3Ak has Lebesgue measure 0 and equals the set of all α ∈ (1, β] for which Sα does not have matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The finer structure of the set of matching parameters α ∈ [1, β] is considered in §§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Before investigating this structure, we show that matching occurs for Sα if and only if it occurs for the corresponding jump transformation Tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The following lemma may be deduced from the general theory of jump transformations, but a proof is included for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 8 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Fix x ∈ [−1, 1] and let j1 < j2 < j3 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' be an enumeration of the set {j ≥ 0 | Sj α(x) ∈ J0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then T k α(x) = Sjk+1 α (x) for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The claim is immediate for k = 1 by (2) and the fact that Sα(J−1 ∪ J1) ⊂ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now suppose the result holds for some k ≥ 1, and let i ∈ {0, 1} be minimal such that Si α(Sjk+1 α (x)) ∈ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By definition, then, jk+1 = jk + i + 1, and T k+1 α x = Tα(Sjk+1 α x) = Si+1 α (Sjk+1 α x) = Sjk+1+1 α x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The matching parameters α ∈ [1, β] for Tα and for Sα coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that Tα has critical points at ±1/β, and note that limx→1/β− Tα(x) = 1 while limx→1/β+ Tα(x) = β(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Due to symmetry, Tα has matching if and only if there are integers M, N > 0 for which T M α (1) = T N α (β(1 − α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Suppose first that Tα has matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then T M α (1) = T N α (β(1 − α)) for some M, N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By (2) and the fact that Sα(1−α) = β(1−α), this implies the existence of some M ′, N ′ > 0 for which SM ′ α (1) = SN ′ α (1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Conversely, suppose Sα has matching with matching index m := m(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 it is clear that Sm α (1) = Sm α (1 − α) ∈ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='8, there are M, N > 0 for which T M α (1) = Sm+1 α (1) = Sm+1 α (1 − α) = Sm α (β(1 − α)) = T N α (β(1 − α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Matching words and intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' When Sα has matching, we call the first m(α) < ∞ digits of the Sα-expansion of 1 the matching word corresponding to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A maximal subinterval of [1, β] on which matching words coincide is called a matching interval corresponding to the common matching word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Here we classify matching words and matching intervals (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' as all matching parameters belong to some matching interval, this gives a complete classification of matching parameters α ∈ [1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='13, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='18 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='19 imply that this also classifies the first m(α) < ∞ digits of the Sα-expansions of 1 − α for Sα with matching and the maximal subintervals of parameters α on which these digits coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=') Note that matching words and intervals for α ∈ [1, β]\\(1, 1 + 1/β3) have been implicitly determined via the cases considered in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For instance, (1 + 1/β2, β] is the matching interval corresponding to the matching word 10, and the Sα-expansion of 1 − α for each α ∈ (1 + 1/β2, β] begins with 0(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Similarly, (1 + 1/β3, 1 + 1/β2) is the matching interval corresponding to the matching word 1001, and the Sα-expansion of 1 − α for each α in this interval begins with 00(−1)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Denote by ≺ the lexicographical ordering on {0, ±1}N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that ≺ may also be defined on the set {0, ±1}∗ of finite words with alphabet −1, 0, 1 by first sending w ∈ {0, ±1}∗ to w0∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let w0 := 00 ≺ w1 := 001 ≺ w2 := 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We say that d ∈ {0, 1}∗ is in admissible block form if d = 10 or d = 1wi1wi2 · · · win(1 − in/2) for some i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , in ∈ {0, 1, 2}, n ≥ 1 with in ̸= 1, and, when n ≥ 2, i1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The collection of all words in admissible block form is denoted B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The condition that a word in admissible block form ends in win(1 − in/2), in ̸= 1, guarantees that the final three digits are either 001 or 010 (recall (5));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' however, not every word ending this way belongs to B: Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' One verifies that d := 1w2w0w1w01 = 10100001001 ∈ B, whereas d′ := 1010001 /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 9 Note that the indices ij for d ∈ B are uniquely determined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' that is, if 1wi1wi2 · · · win(1 − in/2) = 1wj1wj2 · · · wjm(1 − jm/2), then m = n and ik = jk for each 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Define ϕ : B → {0, −1}∗ by ϕ(10) = 01 and for each d ∈ B of the form d = 1wi1wi2 · · · win(1 − in/2), by ϕ(d) := 0w2−i1w2−i2 · · · w2−in(in/2), where w := −w for each w ∈ {0, ±1}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let σ : {0, ±1}N → {0, ±1}N denote the left shift defined by σ((wj)j≥1) = (wj+1)j≥1 for each (wj)j≥1 ∈ {0, ±1}N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' as with the lexicographical ordering, σ is also defined on the set {0, ±1}∗ of finite words by sending w ∈ {0, ±1}∗ to w0∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We remark that for each T ∈ {Sα, Tα, B}, the left shift of the T-expansion of x equals the T-expansion of T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A word d ∈ B satisfies Property M if, for each j ≥ 0, both σj(d) ⪯ d and σj(ϕ(d)) ⪯ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Denote by M ⊂ B the collection of all words d satisfying Property M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We call 10 and 1001 the exceptional words in M and denote by MU := M\\{10, 1001} the collection of unexceptional words in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let d ∈ B be as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then ϕ(d) = 0w0w2w1w20 = 00001001010, and since both σj(d) ⪯ d and σj(ϕ(d)) ⪯ d for all j ≥ 0, we have d ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We shall see that Property M classifies matching words of the maps Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' To show that M contains all matching words we need the following observation, which is not novel, but for which a proof is included for completeness: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Fix α ∈ [1, β] and x, y ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then x < y if and only if (sα,j(x))j≥1 ≺ (sα,j(y))j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Similarly, for x, y ∈ [0, 1], x < y if and only if (bj(x))j≥1 ≺ (bj(y))j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Suppose x, y ∈ [−1, 1] with x < y, and let n := minj≥1{sα,j(x) ̸= sα,j(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We first claim for each 0 ≤ j < n that Sj α(x) < Sj α(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This is true by assumption for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If n = 1, we’re finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume n > 1 and that the claim holds for some j = k − 1 with 0 ≤ k − 1 < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since sα,k(x) = sα,k(y), we have that Sα restricts to a linear function with positive slope on an interval containing Sk−1 α (x) and Sk−1 α (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' But Sk−1 α (x) < Sk−1 α (y) by assumption, so also Sk α(x) < Sk α(y) and the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since sα,n(x) ̸= sα,n(y) and Sn−1 α (x) < Sn−1 α (y), it must be true that sα,n(x) < sα,n(y) and hence (sα,j(x))j≥1 ≺ (sα,j(y))j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now suppose x ≥ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If equality holds, then by uniqueness of Sα-expansions, (sα,j(x))j≥1 = (sα,j(y))j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If the inequality is strict, the argument above applies with x and y interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The proof of the second statement is identical, mutatis mutandis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Suppose for some α ∈ [1, β] that Sα has matching with index m := m(α), and let d := d1 · · · dm denote the corresponding matching word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then d ∈ M, and e := ϕ(d) agrees with the first m digits e1 · · · em of the Sα-expansion of 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From the cases of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, the result holds for α /∈ (1, 1 + 1/β3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' in particular, α ∈ (1 + 1/β2, β] and α ∈ (1 + 1/β3, 1 + 1/β2) correspond to the exceptional words d = 10 and d = 1001, respectively, in M, and ϕ(10) = 01, ϕ(1001) = 0010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now assume α ∈ (1, 1 + 1/β3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that d1 = 1, e1 = 0, and Sα(1) − Sα(1 − α) = (β − α) − β(1 − α) = α/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall from Equation (5) and the discussion preceding it that �dm−2dm−1dm em−2em−1em � ∈ ��001 010 � , �010 001 �� = ��w01 w20 � , �w20 w01 �� , and Sm−3 α (1) − Sm−3 α (1 − α) = α/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The remaining digits �d2d3 · · · dm−3 e2e3 · · · em−3 � 10 are thus determined by edge labels of cycles in the graph of Figure 2 beginning and ending at vertex α/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' There are three possible cycles, whose edge labels give �djdj+1 ejej+1 � = �01 00 � = �w2 w0 � , �djdj+1 ejej+1 � = �00 01 � = �w0 w2 � , and �djdj+1dj+2 ejej+1ej+2 � = �001 001 � = �w1 w1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It follows that d = 1wi1wi2 · · · win(1−in/2) and e1 · · · em = 0w2−i1w2−i2 · · · w2−in(in/2) for some i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , in ∈ {0, 1, 2}, n ≥ 1 and in ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, note from case (v) of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 that d1d2d3d4 = 1010, so i1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus d ∈ B and e = e1 · · · em = ϕ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='12, the facts that Sj α(1), Sj α(1 − α) ∈ [−1, 1] for each j ≥ 0 imply that σj(d), σj(e) ⪯ d for each j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus d ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The previous result states that every matching word belongs to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Before proving the converse (Propo- sitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='16 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='18), we define and investigate properties of the valuation function v : S → R given by the (absolutely) convergent series v((wj)j≥1) := � j≥1 wj/βj, where S ⊂ ZN consists of all sequences (wj)j≥1 whose entries are bounded above and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The valuation function is also defined on the set S∗ ⊂ S of finite words by considering the corresponding finite sum and setting v(ε) = 0 for the empty word ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It is not difficult to check for finite words w, w′ ∈ {0, ±1}∗ with no consecutive nonzero digits that w ≺ w′ if and only if v(w) < v(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If w := w1w2 · · · wk ∈ {0, 1, 2}∗ is ε (in which case we set k = 0) or consists solely of blocks of 01’s and 002’s, then v(w) = 1/β − 1/βk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The case that w = ε is trivial, so suppose w ̸= ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' One easily verifies that v((01)3) = v((002)2) and v(01002) = v(00201).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These observations, together with the fact that for each 1 ≤ j ≤ k, v(w) = v(w1 · · · wj) + (1/βj)v(wj+1 · · · wk), imply that v(w) = � � � � � v((002)k/3), k ≡ 0 (mod 3) v((002)(k−4)/3(01)2), k ≡ 1 (mod 3) v((002)(k−2)/301), k ≡ 2 (mod 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Notice that for any j ≥ 1, v((002)j) = 2 j � i=1 (1/β3)i = 2 · 1/β3 − 1/β3j+3 1 − 1/β3 = 2 · 1 − 1/β3j β3 − 1 = 2 · 1 − 1/β3j 2β = 1/β − 1/β3j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 11 If k ≡ 0 (mod 3), setting j = k/3 gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If k ≡ 1 (mod 3), we compute v(w) = v((002)(k−4)/3(01)2) = v((002)(k−4)/3) + (1/βk−4)v((01)2) = 1/β − 1/βk−3 + (1/βk−4)(1/β2 + 1/β4) = 1/β − 1/βk−3 + 1/βk−2 + 1/βk = 1/β − 1/βk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Similarly, if k ≡ 2 (mod 3), v(w) = v((002)(k−2)/301) = v((002)(k−2)/3) + (1/βk−2)v(01) = 1/β − 1/βk−1 + 1/βk = 1/β − 1/βk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ For equal-length words x, y ∈ {0, ±1}∗, define x+y, x−y ∈ {0, ±1, ±2}∗ where addition and subtraction, respectively, are performed entry-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that w2 − w0 = 01, w0 − w2 = 01, and w1 − w1 = 002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Suppose d satisfies Property M with m := len(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since d is in admissible block form, the definition of e := ϕ(d) implies that d − e = 1w1 for some word w consisting solely of blocks of 01’s and 002’s or w = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='14, we compute v(d − e) = v(1w1) = 1/β + (1/β)(1/β − 1/βm−1) + 1/βm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This proves the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If d ∈ M and e := ϕ(d), then v(d) − v(e) = v(d − e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For d = 10, set Id = (α− d , α+ d ] := (1 + 1/β2, β], and for all other d = d1 · · · dm ∈ M, define Id = (α− d , α+ d ) := � βm + βdm βmv(d) + βdm , βm − β1−dm βmv(d) − β1−dm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (7) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each d ∈ M, Id is a nonempty subinterval of (1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The result is true for d = 10, so assume d ̸= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We first show that Id ̸= ∅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' that βm + βdm βmv(d) + βdm < βm − β1−dm βmv(d) − β1−dm , or (βm + βdm)(βmv(d) − β1−dm) < (βm − β1−dm)(βmv(d) + βdm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Distributing and cancelling terms gives that this is equivalent to βm+dmv(d) − βm+1−dm < βm+dm − βm+1−dmv(d), or v(d) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since d has no consecutive 1’s, one finds that v(d) < v((10)∞) = 1 (see also Lemma 1 of [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Next we show that Id ⊂ (1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The left endpoint of Id is greater than 1 again since v(d) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It remains to show that βm − β1−dm βmv(d) − β1−dm ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that d1 = 1, and if dm = 0, then dm−1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' thus v(d) ≥ 1/β + β1−dm/βm, and βm+1v(d) − β2−dm ≥ βm+1(1/β + β1−dm/βm) − β2−dm > βm − β1−dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Dividing both sides by βmv(d) − β1−dm gives the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ 12 For each u ∈ {0, 1}∗, let ∆(u) denote the cylinder of points x ∈ [0, 1] for which the β-expansion of x begins with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' One finds for each u = u1 · · · un with ujuj+1 = 0, 1 ≤ j < n, that ∆(u) = � [v(u), v(u) + 1/βn), un = 0 [v(u), v(u) + 1/βn+1), un = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (8) The following lemma is needed in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='18 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let d ∈ MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then Bj(1/α− d ) ≤ 1/α− d and Bj(1 − 1/α+ d ) ≤ 1/α+ d for all j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This is a corollary of two technical results (Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2), whose statements and proofs are provided in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The next result—together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='16—states that every word d ∈ M is in fact a matching word, thus completing our classification of matching words as the set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, it states that the interval Id is contained in a matching interval corresponding to the matching word d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For any d ∈ M and α ∈ Id, the Sα-expansions of 1 and 1 − α begin with d and ϕ(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, Sα has matching with matching index m(α) = len(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The result is shown for exceptional words d ∈ {10, 1001} in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, so assume d ∈ MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Suppose the first statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' That Sα has matching with index m(α) = len(d) is implied by the final three digits of d and e (see the discussion surrounding Equation (5)), so we need only prove the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let α ∈ Id, and write d = d1 · · · dm and e := ϕ(d) = e1 · · · em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We must show that dα,1 · · · dα,m = d1 · · · dm and eα,1 · · · eα,m = e1 · · · em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume that �dm−2dm−1dm em−2em−1em � = �001 010 � , and set α0 := 1/v(d) (the case that dm = 0 is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='16 together with the fact that v(d) < 1 imply α− < α0 < α+, where, for ease of notation, α± := α± d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We claim that it suffices to show the following: (i) if α ∈ (α−, α0), then ℓα(1) > m − 1, ℓα(1 − α) = m − 2, b1(1/α) · · · bm(1/α) = d1 · · · dm, and b1(1 − 1/α) · · · bm−2(1 − 1/α) = e1 · · · em−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (ii) if α ∈ (α0, α+), then ℓα(1) = m − 1, ℓα(1 − α) > m − 2, b1(1/α) · · · bm−1(1/α) = d1 · · · dm−1, and b1(1 − 1/α) · · · bm(1 − 1/α) = e1 · · · em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' and (iii) if α = α0, then ℓα(1) = m − 1, ℓα(1 − α) = m − 2, b1(1/α) · · · bm−1(1/α) = d1 · · · dm−1, b1(1 − 1/α) · · · bm−2(1 − 1/α) = e1 · · · em−2, and Bm−1(1/α) = Bm−2(1 − 1/α) = 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Indeed, suppose (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 implies dα,1 · · · dα,m = d1 · · · dm and eα,1 · · · eα,m−2 = e1 · · · em−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since ℓα(1 − α) = m − 2, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 gives Sm−2 α (1 − α) ∈ [−α/β, −1/β), so eα,m−1 = −1 and eα,m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In case (ii), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 again gives dα,1 · · · dα,m−1 = d1 · · · dm−1 13 and eα,1 · · · eα,m−1 = e1 · · · em−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, ℓα(1) = m − 1 implies Sm−1 α (1) ∈ (1/β, α, β] and hence dα,m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since eα,m−1 = em−1 = −1, it follows that eα,m = 0 = em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In (iii), we have dα,1 · · · dα,m−1 = d1 · · · dm−1 and eα,1 · · · eα,m−2 = e1 · · · em−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 gives Sm−1 α (1) = −Sm−2 α (1 − α) = α/β, so dα,m = eα,m−1 = 1 and eα,m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6, (i), (ii) and (iii) are implied by showing: (a) 1/Id ⊊ ∆(d1 · · · dm−1) and 1 − 1/Id ⊊ ∆(e1 · · · em−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (b) Bj(1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m−1, and Bj(1−1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (c) if α ∈ (α−, α0), then Bm−1(1/α) > 1/β and Bm−2(1 − 1/α) ∈ (1/βα, 1/β];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (d) if α ∈ (α0, α+), then Bm−1(1/α) ∈ (1/βα, 1/β] and Bm−2(1 − 1/α) > 1/β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' and (e) if α = α0, then Bm−1(1/α) = Bm−2(1 − 1/α) = 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We prove each of (a), (b), (c), (d) and (e): (a) The first inclusion is equivalent to v(d1 · · · dm−1) < 1/α+ < 1/α− < v(d1 · · · dm−1) + 1/βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (9) Note that v(d1 · · · dm−1) < 1/α+ if and only if v(d) − 1/βm < βmv(d) − 1 βm − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Multiplying both sides by βm−1, cancelling and rearranging terms, this is equivalent to v(d) > 1/βm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This latter inequality holds since v(d) ≥ v(d1) = 1/β and m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Next, 1/α− < v(d1 · · · dm−1) + 1/βm−1 if and only if βmv(d) + β βm + β < v(d) − 1/βm + 1/βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Using the fact that 1/βm−1 = 1/βm+1/βm+1 and multiplying both sides by βm+β, this is equivalent to βmv(d) + β < (βm + β)(v(d) + 1/βm+1), or βmv(d) + β < βmv(d) + 1/β + βv(d) + 1/βm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Simplifying, this is equivalent to showing 1 < βv(d) + 1/βm, which again holds since v(d) ≥ 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus 1/Id ⊊ ∆(d1 · · · dm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The second inclusion is equivalent to v(e1 · · · em−2) < 1 − 1/α− < 1 − 1/α+ < v(e1 · · · em−2) + 1/βm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now v(e1 · · · em−2) < 1 − 1/α− if and only if 1/α− < 1 − (v(e) − 1/βm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='15, the fact that v(e) = −v(e) and (9), 1 − (v(e) − 1/βm−1) = v(d) + 1/βm−1 > v(d1 · · · dm−1) + 1/βm−1 > 1/α−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lastly, 1 − 1/α+ < v(e1 · · · em−2) + 1/βm−2 if and only if 1 − 1/α+ < v(e) − 1/βm−1 + 1/βm−2, or v(d) < 1/α+ + 1/βm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From (9), we find v(d) − 1/βm = v(d1 · · · dm−1) < 1/α+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus 1 − 1/Id ⊊ ∆(e1 · · · em−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (b) Fix 0 ≤ j < m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If dj+1 = 1, then part (a) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='12 imply that Bj(1/α) > Bj(1/α+) ≥ 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now suppose dj+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By (a), Bj(1/α−) ∈ (1/βα−, 1/β] if and only if Bj+1(1/α−) ∈ (1/α−, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='17 thus implies Bj(1/α−) /∈ (1/βα−, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Equation (6), it also holds for each x ∈ ∆(d1 · · · dm−1) that Bj(x) /∈ (x/β, 1/β] if and only if βj(x − v(d1 · · · dj)) ≤ x/β, 14 or x ≤ βjv(d1 · · · dj) βj − 1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since 1/α, 1/α− ∈ ∆(d1 · · · dm−1) and Bj(1/α−) /∈ (1/βα−, 1/β], we have 1/α < 1/α− ≤ βjv(d1 · · · dj) βj − 1/β , which implies Bj(1/α) /∈ (1/βα, 1/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus Bj(1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The proof that Bj(1 − 1/α) /∈ (1/βα, 1/β] for each 0 ≤ j < m − 2 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (c) Suppose α ∈ (α−, α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From Equation (6) and part (a), we have for each x ∈ 1/Id that Bm−1(x) = βm−1(x − v(d1 · · · dm−1)) (10) = βm−1(x − (v(d) − 1/βm)) Since 1/α > 1/α0 = v(d), we have Bm−1(1/α) > 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Also from Equation (6), part (a) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='15, for each x ∈ 1/Id, Bm−2(1 − x) = βm−2(1 − x − v(e1 · · · em−2)) (11) = βm−2(1 − x + v(e) + 1/βm−1) = βm−2(−x + v(d) + 1/βm−1) = −βm−2x + βm−2v(d) + 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Hence Bm−2(1 − 1/α) < Bm−2(1 − 1/α0) = 1/β, and Bm−2(1 − 1/α) > 1/βα if and only if βm−2v(d) + 1/β βm−2 + 1/β > 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' But the left hand side equals 1/α−, so the inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (d) Suppose α ∈ (α0, α+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From Equation (10), 1/α < 1/α0 = v(d) implies Bm−1(1/α) < 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, Bm−1(1/α) > 1/βα if and only if 1/α > βm−1v(d) − 1/β βm−1 − 1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The right-hand side equals 1/α+, and α < α+ by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We also find from Equation (11) that Bm−2(1 − 1/α) = βm−2(v(d) − 1/α) + 1/β > 1/β since 1/α < 1/α0 = v(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (e) This again follows from Equations (10) and (11), setting x = 1/α0 = v(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The following proposition states that the interval Id contains the matching intervals corresponding to the matching word d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='18, this characterises matching intervals as the collection {Id}d∈M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If Sα has matching with m(α) = m, then α ∈ Id, where d = d1 · · · dm is beginning of the Sα-expansion of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='13, d ∈ M, so Id is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The result holds for m ≤ 2 by the cases in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, so assume m > 2 and let e = e1 · · · em denote the beginning of the Sα-expansion of 1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall from Equation (5) that �dm−2dm−1dm em−2em−1em � ∈ ��010 001 � , �001 010 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume dm = 0 (the other case is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 and the final digits of d and e imply that either (i) Sm−2 α (1) ∈ (1/β, α/β] or (ii) Sm−1 α (1 − α) ∈ [−α/β, −1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 15 It suffices to show that both (i) and (ii) imply α ∈ Id = � βm + 1 βmv(d) + 1, βm − β βmv(d) − β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (i) Equation (3) gives Sm−2 α (1) = βm−2(1 − αv(d1 · · · dm−2)) ∈ (1/β, α/β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that v(d1 · · · dm−2) = v(d) − 1/βm−1, so 1 − α(v(d) − 1/βm−1) ∈ (1/βm−1, α/βm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now 1 − α(v(d) − 1/βm−1) > 1/βm−1 implies α < 1 − 1/βm−1 v(d) − 1/βm−1 = βm − β βmv(d) − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, 1 − α(v(d) − 1/βm−1) ≤ α/βm−1 gives 1 ≤ αv(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus we have α ∈ � 1 v(d), βm − β βmv(d) − β � , and it suffices to show that βm + 1 βmv(d) + 1 < 1 v(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' But this is true since v(d) < v((10)∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (ii) Again from Equation (3), Sm−1 α (1 − α) = βm−1(1 − α(1 + v(e1 · · · em−1))) ∈ [−α/β, −1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The assumption that em = −1 together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='15 give 1 + v(e1 · · · em−1) = 1 + v(e) + 1/βm = v(d) + 1/βm, so 1 − α(v(d) + 1/βm) ∈ [−α/βm, −1/βm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now 1 − α(v(d) + 1/βm) ≥ −α/βm implies 1 ≥ αv(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Furthermore, 1 − α(v(d) + 1/βm) < −1/βm gives α > 1 + 1/βm v(d) + 1/βm = βm + 1 βmv(d) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Hence α ∈ � βm + 1 βmv(d) + 1, 1 v(d) � , and it suffices to show 1 v(d) < βm − β βmv(d) − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This is true again since v(d) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The implications of Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='13, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='16, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='18 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='19 are summarised in the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The sets M and {Id}d∈M classify the matching words and intervals, respectively, of the maps Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 16 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The results of this subsection also imply that ϕ(M) classifies the first m(α) < ∞ digits of the Sα-expansions of 1 − α for matching parameters α ∈ [1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, the intervals Id in {Id}d∈M = {Iϕ−1(e)}e∈ϕ(M) classify the maximal subintervals of matching parameters α for which these first m(α) digits coincide (and equal e = ϕ(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' While not needed for our purposes, we briefly mention that the sets M (or ϕ(M)) and {Id}d∈M also give rise to classifications of the Tα-expansions of 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' β(1 − α)) before matching and the maximal intervals of parameters α on which these expansions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, if d ∈ M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' e := ϕ(d) ∈ ϕ(M)), then the corresponding Tα-word d′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' e′) ‘forgets’ each non-terminal 0 which immediately follows a 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' −1, and e′ also forgets the initial 0 of e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The matching intervals Id are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For instance, d = 10100001 and e = ϕ(d) = 00001010 give rise to the words d′ = 110001 and e′ = 000110 for Tα, and each of these words corresponds to the matching interval Id = � β8+β β7+β5+β2 , β8−1 β7+β5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Cascades of matching intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Here it is shown that each unexceptional matching interval Id, d ∈ MU, generates a whole ‘cascade’ of unexceptional matching intervals with adjacent endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Define ψ : MU → {0, 1}∗, where for d = d1 · · · dm ∈ MU and e := ϕ(d) = e1 · · · em, ψ(d) = � de, dm = 0 de2 · · · em, dm = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall the definition of the matching interval Id = (α− d , α+ d ) from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The map ψ preserves Property M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' ψ(MU) ⊂ MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, α− d = α+ ψ(d) for each d ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let d = d1 · · · dm ∈ MU, and assume dm = 0 (the other case is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We first show α− d = α+ ψ(d), assuming ψ(MU) ⊂ MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We compute α+ ψ(d) = β2m − 1 β2mv(de) − 1 = (βm + 1)(βm − 1) β2m(v(d) − (1/βm)v(e)) − 1 = (βm + 1)(βm − 1) β2mv(d) − βm(v(d) − 1) − 1 = (βm + 1)(βm − 1) (βmv(d) + 1)(βm − 1) = βm + 1 βmv(d) + 1 = α− d as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now we prove that d′ := ψ(d) ∈ MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Clearly d′ /∈ {10, 1001}, so we need only show d′ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Write d = 1wi1 · · · win0 with in = 2 and e = ϕ(d) = 0w2−i1 · · · w2−in1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then d′ = de = 1wi1 · · · win00w2−i1 · · · w2−in1 = 1wi1 · · · winw0w2−i1 · · · w2−in1, so d′ ∈ B is in admissible block form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' To prove d′ ∈ M, it remains to show for each j ≥ 0 that (i) σj(d′) ⪯ d′ and (ii) σj(ϕ(d′)) ⪯ d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (Recall that d ∈ M implies the analogous inequalities hold for d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=') 17 (i) If j ≥ m, then σj(d′) = σj(de) = σj−m(e) ⪯ d ⪯ d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume j < m, and suppose for the sake of contradiction that σj(d′) ≻ d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since d′ begins with 1, so does σj(d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus either σj(d′) = 1wiℓ · · · winw0w2−i1 · · · w2−in1 for some 1 < ℓ ≤ n, or σj(d′) = 1w0w2−i1 · · · w2−in1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since w0 ≺ w2 = wi1, the second case is impossible and we must have 1wiℓ · · · winw0w2−i1 · · · w2−in1 ≻ 1wi1 · · · winw0w2−i1 · · · w2−in1 for some ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since σj(d) ⪯ d, it follows that 1wiℓ · · · win = 1wi1 · · · win−ℓ+1 and thus w0w2−i1 · · · w2−in1 ≻ win−ℓ+2 · · · winw0w2−i1 · · · w2−in1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then either there is some 1 ≤ p ≤ ℓ − 3 for which (0, 2 − i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , 2 − ip−1) = (in−ℓ+2, in−ℓ+3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , in+p−ℓ+1) and 2 − ip > in+p−ℓ+2, or (0, 2 − i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , 2 − iℓ−2) = (in−ℓ+2, in−ℓ+3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In the first case, (2 − in−ℓ+2, 2 − in−ℓ+3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , 2 − in+p−ℓ+1) = (2, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , ip−1) and 2 − in+p−ℓ+2 > ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus there exists some k ≥ 0 for which σk(e) = 1w2−in−ℓ+3 · · · w2−in+p−ℓ+1w2−in+p−ℓ+2 · · · w2−in1 ≻ 1wi1 · · · wip−1wip · · · win0 = d, contradicting the fact that d ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In the second case, (2 − in−ℓ+2, 2 − in−ℓ+3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , 2 − in) = (2, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , iℓ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since in = 2 implies iℓ−2 = 0, there is again some k ≥ 0 for which σk(e) = 1w2−in−ℓ+3 · · · w2−in−1w2−in1 = 1w2−in−ℓ+3 · · · w2−in−1w1 ≻ 1wi1 · · · wiℓ−3wiℓ−2 · · · win0 = d, contradicting d ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (ii) Set e′ := ϕ(d′) = e1 · · · em, and recall that dm = 0 implies em = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then e′ = 0w2−i1 · · · w2−inw2wi1 · · · win0 = e1 · · · em−10d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If j < m − 1, then σj(e′) = ej+1 · · · em−10d ≺ ej+1 · · · em = σj(e) ⪯ d ⪯ d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If j = m − 1, then σj(e′) = 0d ≺ d′, and if j ≥ m, then σj(e′) = σj−m(d) ⪯ d ⪯ d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' This concludes the proof that d′ = ψ(d) ∈ M and thus ψ(MU) ⊂ MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Invariant measures and frequencies of digits As noted above, our main interest in matching arises from results of [17] which provide explicit expressions for the densities of absolutely continuous invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These densities depend on the orbits of the left and right limits at critical points and are in general infinite sums of (finite) step functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' however, the infinite sum becomes finite when either matching or a Markov partition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These observations are used in this section to obtain explicit invariant measures να and µα for the maps Sα and Tα, respectively, and asymptotic relative frequencies of digits occurring in their respective generic expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These measures and frequencies are used in the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that B(x) := βx (mod 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It is well known that h(x) := � 5+3 √ 5 10 , x ∈ [0, 1/β) 5+ √ 5 10 , x ∈ [1/β, 1] is the density of a unique, ergodic, B-invariant probability measure which is equivalent to Lebesgue measure λ ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Birkhoff’s ergodic theorem, the frequency of 0 in λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' β-expansion is � [0,1/β) hdλ = (5+ √ 5)/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' When α = 1, the map Sα = S1 restricts on [0, 1]\\{1/β} to B and on [−1, 0]\\{−1/β} to −B(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since S1 is invariant on ±[0, 1], we find that the frequency of 0 in λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' S1-expansion is also (5 + √ 5)/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Define f1 : [−1, 1] → [−1, 1] by f1(x) = h(|x|)/2, and recall the definitions of the subintervals Ji ⊂ [−1, 1], i ∈ {−1, 0, 1} from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note, then, that the measure ν1 defined on Lebesgue-measurable A ⊂ [−1, 1] by ν1(A) = � A f1dλ satisfies ν1(J0) := (5 + √ 5)/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A similar analysis (with Lebesgue measure) reveals that the frequency of 0 in λ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' T1-expansion is 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Setting µ1 := λ/2 as normalised Lebesgue measure gives µ1(J0) = 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In what follows we consider α ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let α ∈ (1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Following a procedure completely analogous to that in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 of [13], results of [17] imply that the collection of absolutely continuous Sα-invariant measures forms a one real- dimensional linear space and thus there is a unique—and hence ergodic—absolutely continuous invariant probability measure να.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, its corresponding probability density is given explicitly by fα(x) := 1 C � t≥0 1 βt+1 � 1[−1,St α(α−1))(x) − 1[−1,St α(−1))(x) + 1[−1,St α(1))(x) − 1[−1,St α(1−α))(x) � , where C ∈ R is some normalising constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Symmetry of Sα together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 allow us to rewrite fα(x) as fα(x) = 1 C � t≥0 1 βt+1 � 1[Stα(−1),Stα(α−1))(x) + 1[Stα(1−α),Stα(1))(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (12) Note that fα is bounded away from 0 on [−1, 1), so να is in fact equivalent to Lebesgue measure λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Also observe that when matching (or a Markov partition) occurs, the summation becomes a finite sum and fα(x) is a (finite) step function (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The measure να can now be used to obtain a unique, absolutely continuous Tα-invariant measure µα = � gαdλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each α ∈ (1, β], define a probability measure µα(A) := να � S−1 α (A) ∩ J0 � να(J0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (13) on [−1, 1], where A ⊂ [−1, 1] is Lebesgue-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that S−1 α (A) ∩ J0 = 1 β A, so µα may also be written µα(A) = να( 1 β A)/να(J0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The measure µα is the unique—hence ergodic—invariant probability measure for Tα which is absolutely continuous with respect to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, µα is equivalent to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since Tα is an expanding, piecewise C2 monotone map, results of [19] imply the existence of an invariant probability measure ρα for Tα which is absolutely continuous with respect to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let J±1 := J−1∪J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' As Tα is a jump transformation for Sα, the measure ρα induces an Sα-invariant measure defined by ˜ρα(A) := ρα(A) + ρα � S−1 α (A) ∩ J±1 � (14) 19 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The invariant densities fα for Sα (red) and gα for Tα (blue) with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='16 (left), α = 1/v(1010) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='17082 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (center) and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 of [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that for any A ⊂ J±1 we have S−1 α (A) ⊂ J0, so (14) gives ˜ρα(A) = ρα(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then for any measurable A ⊂ [−1, 1], ˜ρα � S−1 α (A) ∩ J±1 � = ρα � S−1 α (A) ∩ J±1 � and (14) gives ρα(A) = ˜ρα(A) − ˜ρα � S−1 α (A) ∩ J±1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since ˜ρα is Sα-invariant, the previous line may be rewritten ρα(A) = ˜ρα(S−1 α (A)) − ˜ρα � S−1 α (A) ∩ J±1 � = ˜ρα(S−1 α (A) ∩ J0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that να is the unique invariant, absolutely continuous probability measure for Sα, so ˜ρα = cνα for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus ρα(A) = cνα � S−1 α (A) ∩ J0 � , and setting A = [−1, 1] gives c = 1/να(J0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Hence ρα = µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' That µα is equivalent to Lebesgue measure λ follows immediately from the fact that να is equivalent to λ and the observation above that µα(A) = να( 1 β A)/να(J0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 asserts the existence of a unique, absolutely continuous Tα-invariant probability measure µα which is in fact equivalent to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It remains to show that for fixed d ∈ M, the density gα of each µα, α ∈ Id, is a step function with at most the same, finite number of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Using a change of variables, one finds that µα(A) = να( 1 β A) να(J0) = 1 να(J0) � 1 β A fα(x)dλ(x) = 1 βνα(J0) � A fα(x/β)dλ(x), so gα(x) = fα(x/β) βνα(J0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since, by (12), fα is a linear combination of at most 2m(α) indicator functions and m(α) is constant on Id, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The number of jumps of the invariant densities fα and gα for Sα and Tα, respectively, are non-constant on matching intervals Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Figure 3 shows these densities for three values of α in the matching interval Id ≈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='14589 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='23606 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' ) with d = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that the number of jumps is fewer for α = 1/v(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' One can show that this phenomenon generalises to all matching intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' in fact, for each d ∈ M, the number of jumps of fα and gα, respectively, are constant for all but finitely many α ∈ Id, and the number of jumps decreases for α = 1/v(d) ∈ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='0Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The frequency functions fS(α) (red) and fT (α) (blue) plotted on all matching intervals Id with len(d) ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The visible plateaux correspond to the interval [1/2+1/β, 1+ 1/β2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Frequencies of digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We are now in a position to determine the frequencies of digits in generic Sα- and Tα-expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Define fS, fT : [1, β] → [0, 1] by fS(α) := να(J0) and fT (α) := µα(J0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For α ̸= 1, Birkhoff’s ergodic theorem—together with the equivalence of the ergodic measures να and µα with Lebesgue measure λ—implies that the asymptotic frequencies lim n→∞ 1 n n−1 � i=0 1J0(Si α(x)) and lim n→∞ 1 n n−1 � i=0 1J0(T i α(x)) of the digit 0 in Lebesgue-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Sα- and Tα-expansion are given by fS(α) and fT (α), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Indeed, with the discussion and notation given at the beginning of §3, fS(1) and fT (1) also give the generic asymptotic frequencies of the digit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note, too, that the frequencies of the digits ±1 are readily obtained from the frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, set J±1 := J−1 ∪ J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Using (13) and the Sα-invariance of να, one has for any measurable A ⊂ [−1, 1], µα(A) = να(S−1 α (A)) − να(S−1 α (A) ∩ J±1) να(J0) = να(A) − να(S−1 α (A) ∩ J±1) να(J0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Setting A = J0 and using the fact that S−1 α (J0) ∩ J±1 = J±1, we find µα(J0) = να(J0) − να(J±1) να(J0) = να(J0) − (1 − να(J0)) να(J0) or fT (α) = 2 − 1 fS(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (15) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The frequency functions fS and fT are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Arguments completely analogous to those in §4 of [13] give that fS is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Continuity of fT is immediate from 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ The remainder of this subsection is devoted to finding—for matching parameters α—an explicit expression for fS(α) in terms of α and its corresponding matching word d (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Density of matching parameters in [1, β], continuity of fS and equation (15) then allow us to determine fS(α) and fT (α) for any α ∈ [1, β] as limits of these explicit expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' These expressions are then used in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3 to determine the maximal 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6frequency of the digit 0 occurring in generic Sα- and Tα-expansions, and it is shown that these maximal values are attained for α in the interval [1/2 + 1/β, 1 + 1/β2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume that α ∈ Id, d ∈ M, with matching index m := m(α) < ∞, and recall the density fα from equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We first find an expression for the normalising constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By symmetry of Sα, 1 = να([−1, 1]) = � 1 −1 fα(x)dλ(x) = 2 C m−1 � t=0 � 1 −1 1 βt+1 1[Stα(1−α),Stα(1))(x)dλ(x) = 2 C m−1 � t=0 1 βt+1 � St α(1) − St α(1 − α) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Assume α < 1 + 1/β2 and write d = d1 · · · dm = 1wi1 · · · win(1 − in/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each i ∈ 0, 1, 2, let ℓ(i) ∈ {2, 3} denote the length of the block wi—explicitly, ℓ(0) = ℓ(2) = 2 and ℓ(1) = 3—and let p := pd : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , m−3} be defined by p(k) = 1+�k−1 j=1 ℓ(ij) so that σp(k)(d) = wik · · · win(1−in/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall from Figure 2 that S0 α(1)−S0 α(1−α) = α, Sm−1 α (1)−Sm−1 α (1−α) = α/β, and that the remaining differences St α(1)−St α(1−α) are determined by cycles of length two or three beginning at vertex α/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, if ik ∈ {0, 2}, then Sp(k) α (1)−Sp(k) α (1−α) = α/β and Sp(k)+1 α (1)−Sp(k)+1 α (1−α) = α give a cycle of length two, while if ik = 1, Sp(k) α (1) − Sp(k) α (1 − α) = α/β, Sp(k)+1 α (1) − Sp(k)+1 α (1 − α) = α and Sp(k)+2 α (1) − Sp(k)+2 α (1 − α) = βα give a cycle of length three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We find for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , n} that p(k)+ℓ(ik)−1 � t=p(k) 1 βt+1 � St α(1) − St α(1 − α) � = ℓ(ik) βp(k)+2 α, and thus 1 = 2 C m−1 � t=0 1 βt+1 � St α(1) − St α(1 − α) � = 2 C � �α β + n � k=1 p(k)+ℓ(ik)−1 � t=p(k) 1 βt+1 � St α(1) − St α(1 − α) � + α βm+1 � � = 2α C � 1 β + n � k=1 ℓ(ik) βp(k)+2 + 1 βm+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (16) Note that (16) also holds for α > 1 + 1/β2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' d = 10) with the summation over k set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Define a substitution ξ : {w0, w1, w2} → {02, 030} by ξ(w0) = ξ(w2) = 02 and ξ(w1) = 030, and let Ξ : M → {0, 1, 2, 3}∗ be given by Ξ(d) = 101 if d = 10, and Ξ(d) = 1ξ(wi1) · · · ξ(win)01 if d = 1wi1 · · · win(1 − in/2) ∈ M\\{10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The left- and right-most sides of (16) may be written more succinctly as 1 = 2α C v(Ξ(d)), and thus C = 2αv(Ξ(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 22 Having found C, we are now in a position to determine fS(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Again by symmetry of Sα, fS(α) = να(J0) = 1 − να(J−1) − να(J1) = 1 − � −1/β −1 fα(x)dλ(x) − � 1 1/β fα(x)dλ(x) = 1 − 2 C m−1 � t=0 �� −1/β −1 1 βt+1 1[Stα(1−α),Stα(1))(x)dλ(x) + � 1 1/β 1 βt+1 1[Stα(1−α),Stα(1))(x)dλ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Write e := ϕ(d) = e1 · · · em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1, St α(1) /∈ J−1 and St α(1 − α) /∈ J1 for t < m, the previous line may be rewritten as fS(α) = 1 − 2 C � � � � � 0≤t≤m−1 et+1=−1 1 βt+1 (−1/β − St α(1 − α)) + � 0≤t≤m−1 dt+1=1 1 βt+1 (St α(1) − 1/β) � � � � = 1 − 2 C � � � � � 0≤t≤m−1 dt+1=1 1 βt+1 St α(1) − � 0≤t≤m−1 et+1=−1 1 βt+1 St α(1 − α) − 1/β � � � � , where we have used Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='15 together with the facts that � 0≤t≤m−1 et+1=−1 1/βt+1 = −v(e) and � 0≤t≤m−1 dt+1=1 1/βt+1 = v(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let d0 1 = e0 1 = ε be the empty word, and for 1 ≤ t ≤ m − 1 set dt 1 := d1 · · · dt and et 1 := e1 · · · et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each 0 ≤ t ≤ m − 1, equation (3) gives St α(1) = βt(1 − αv(dt 1)) and St α(1 − α) = βt(1 − α − αv(et 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Setting n(d) := #{1 ≤ j ≤ m | dj = 1} − #{1 ≤ j ≤ m | ej = −1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (17) the frequency function may be written as fS(α) = 1 − 2 C � � � � � 0≤t≤m−1 dt+1=1 1 βt+1 βt(1 − αv(dt 1)) − � 0≤t≤m−1 et+1=−1 1 βt+1 βt(1 − α − αv(et 1)) − 1/β � � � � = 1 − 2 βC � � � � � 0≤t≤m−1 dt+1=1 (1 − αv(dt 1)) − � 0≤t≤m−1 et+1=−1 (1 − α − αv(et 1)) − 1 � � � � = 1 − 2 βC � � � �n(d) − α � � � � � 0≤t≤m−1 dt+1=1 v(dt 1) − � 0≤t≤m−1 et+1=−1 (1 + v(et 1)) � � � � − 1 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Letting Kd := � 0≤t≤m−1 dt+1=1 v(dt 1) − � 0≤t≤m−1 et+1=−1 (1 + v(et 1)) and recalling that C = 2αv(Ξ(d)), we find fS(α) = 1 − 1 βv(Ξ(d)) �n(d) − 1 α − Kd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (18) 23 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let d = 1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then e = 0010, so n(d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, v(Ξ(d)) = v(10201) = 1 β + 2 β3 + 1 β5 and Kd = v(ε) + v(100) − (1 − v(00)) = − 1 β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus for all α ∈ I1001, fS(α) = 1 − 1 β3(1/β + 2/β3 + 1/β5) = 4/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' A similar calculation with d = 1010 reveals that fS(α) = 4/5 also for all α ∈ I1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Before turning toward the maximal frequency of the digit 0, we give an alternate expression for Kd which will be helpful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that the first summation in the definition of Kd may be rewritten as the sum of all v(dt 1), 1 ≤ t ≤ m, for which dt=1, excluding the greatest such index t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The second sum may be similarly rewritten (though an extra term 1 appears from the first non-zero summand of the original sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Now suppose d ̸= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recalling that {dm−2dm−1dm, em−2em−1em} = {001, 010}, we have Kd = � 1≤t≤m−3 dt=1 v(dt 1) − � � �1 + � 1≤t≤m−3 et=−1 (1 + v(et 1)) � � � = v(1) + � 1≤k≤n−1 ik∈{1,2} v(1wi1 · · · wik) − � � � �1 + � 1≤k≤n−1 2−ik∈{1,2} (1 − v(0w2−i1 · · · w2−ik)) � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that p(k + 1), 1 ≤ k ≤ n − 1, gives the power for which σp(k+1)(d) = wik+1 · · · win(1 − in/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' in particular, p(k + 1) equals the length of 1wi1 · · · wik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='14, v(1wi1 · · · wik) + v(0w2−i1 · · · w2−ik) = 1 β + 1 β � 1 β − 1 βp(k+1) � = 1 − 1/βp(k+1)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then Kd = 1 β + � 1≤k≤n−1 ik∈{1,2} v(1wi1 · · · wik) − � � � �1 + � 1≤k≤n−1 2−ik∈{1,2} � v(1wi1 · · · wik) + 1/βp(k+1)+1� � � � � = − 1 β2 + � 1≤k≤n−1 ik=2 v(1wi1 · · · wik) − � 1≤k≤n−1 ik=0 v(1wi1 · · · wik) − � 1≤k≤n−1 2−ik∈{1,2} 1/βp(k+1)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The latter summation equals � 1≤k≤n−1 2−ik∈{1,2} 1/βp(k+1)+1 = 1 β v(0w2−i1 · · · w2−in−1) = 1 β � v(e) − 1 βm−3 v(w2−inin/2) � = 1 β � 1 − v(d) − 1 βm−3 v(w2−inin/2) � = 1 β � 1 − v(d1 · · · dm−3) − 1 βm−3 v(011) � = 1 β − 1 β v(d1 · · · dm−3) − 1 βm−1 , 24 and thus for d ∈ M\\{10}, Kd = −1 + � 1≤k≤n−1 ik=2 v(1wi1 · · · wik) − � 1≤k≤n−1 ik=0 v(1wi1 · · · wik) + 1 β v(d1 · · · dm−3) + 1 βm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (19) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Maximal frequency of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Here we prove that the frequency functions fS and fT attain their maximums on the (maximal) interval [1/2 + 1/β, 1 + 1/β2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We first need some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that by (18), on the matching interval Id the frequency function fS is strictly increasing with α for n(d) > 1, strictly decreasing for n(d) < 1 and constant for n(d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' By (15), the same monotonicity conditions hold for fT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The first of our preliminary results states that fS (and hence fT ) is constant on ‘cascade’ intervals: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' For each d ∈ MU, we have n(ψ(d)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In particular, for each d ∈ MU, the frequency function fS is constant on [limn→∞ α− ψn(d), α− d ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' It suffices to prove the first statement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' the second follows immediately from this, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='23 and continuity of fS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Write d = d1 · · · dm = 1wi1 · · · win(1 − in/2) and e := ϕ(d) = e1 · · · em = 0w2−i1 · · · w2−in(in/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Observe that d′ := ψ(d) = � de, dm = 0 de2 · · · em, dm = 1 = � 1wi1 · · · win00w2−i1 · · · w2−in(in/2), dm = 0 1wi1 · · · win−1001w2−i1 · · · w2−in(in/2), dm = 1 = � 1wi1 · · · winw0w2−i1 · · · w2−in(in/2), dm = 0 1wi1 · · · win−1w1w2−i1 · · · w2−in(in/2), dm = 1 , so e′ := ϕ(d′) = � 0w2−i1 · · · w2−inw2wi1 · · · win(1 − in/2), dm = 0 0w2−i1 · · · w2−in−1w1wi1 · · · win(1 − in/2), dm = 1 = � e1 · · · em−10d, dm = 0 e1 · · · em−20d, dm = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Recall that if dm = 0, then em = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' In this case d′ has exactly one more digit 1 than does e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If dm = 1, then em−1em = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Since e1 = 0, we see that in this case, too, d′ has exactly one more digit 1 than does e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus in both cases n(d′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' □ We make note here of some computations which will be useful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let c, ℓ ∈ Z with ℓ ≥ 0: v((0c)ℓ) = c ℓ � j=1 1/β2j = c β2 · 1 − 1/β2ℓ 1 − 1/β2 = c β (1 − 1/β2ℓ) (20) v((00c)ℓ) = c ℓ � j=1 1/β3j = c β3 · 1 − 1/β3ℓ 1 − 1/β3 = c 2β (1 − 1/β3ℓ) (21) v((0c0)ℓ) = βv((00c)ℓ) = c 2(1 − 1/β3ℓ) (22) v((000c)ℓ) = c ℓ � j=1 1/β4j = c β4 1 − 1/β4ℓ 1 − 1/β4 = c β(β2 + 1)(1 − 1/β4ℓ) (23) v((0c00)ℓ) = β2v((000c)ℓ) = cβ β2 + 1(1 − 1/β4ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' (24) 25 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' If α ∈ Id for some d ∈ M with n(d) = 1, then fS(α) ≤ 4/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, equality holds if and only if d ≺ 1(w2w0)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Note that n(10) = 0, so we may assume d ≻ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' That fS(α) = 4/5 for all α ∈ I1010 ∪ I1001 was shown in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Thus we may assume that d ≻ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Write d = d1 · · · dm = 1wi1 · · · win(1 − in/2) = 1X1Y1 · · · XtYtwin(1 − in/2), where each Xs and Ys, 1 ≤ s ≤ t, consists solely of w2i’s and w1’s, respectively, and each Xs, Ys ̸= ε except possibly Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let ℓ2s−1 := 1 2len(Xs) and ℓ2s := 1 3len(Ys) denote the number of blocks wi in Xs and Ys, respectively, and set ℓj := 0 for j > 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Analogous to the function p = pd defined in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2, set p1 := 1 and for each s ≥ 1, let p2s := p2s−1 + 2ℓ2s−1 and p2s+1 := p2s + 3ℓ2s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' note, then, that σp2s−1(d) = XsYs · · · XtYtwin(1 − in/2) and σp2s(d) = YsXs+1 · · · XtYtwin(1 − in/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Let k2s−1, k2s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' , n} be the indices for which σp2s−1(d) = wik2s−1 · · · win−1win(1 − in/2) and σp2s(d) = wik2s · · · win−1win(1 − in/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Using (20) and (22), we compute v(Ξ(d)) = v(1(02)ℓ1(030)ℓ2 · · · (02)ℓ2t−1(030)ℓ2t0201) = 1 β + t � s=1 � 1 βp2s−1 v((02)ℓ2s−1) + 1 βp2s v((030)ℓ2s) � + 1 βm−3 v(0201) = 1 β + t � s=1 � 2 βp2s−1+1 (1 − 1/β2ℓ2s−1) + 3 2βp2s (1 − 1/β3ℓ2s) � + 1 βm−3 (2/β2 + 1/β4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Moreover, (21) gives v(d1 · · · dm−3) = 1 β + t � s=1 � 1 βp2s−1 v(Xs) + 1 βp2s v(Ys) � = 1 β + t � s=1 � 1 βp2s−1 v(Xs) + 1 βp2s v((001)ℓ2s) � = 1 β + t � s=1 � 1 βp2s−1 v(Xs) + 1 2βp2s+1 (1 − 1/β3ℓ2s) � , so equation (19) becomes Kd = − 1 β + � 1≤k≤n−1 ik=2 v(1wi1 · · · wik) − � 1≤k≤n−1 ik=0 v(1wi1 · · · wik) + t � s=1 � 1 βp2s−1+1 v(Xs) + 1 2βp2s+2 (1 − 1/β3ℓ2s) � + 1 βm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='Then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βv(Ξ(d)) + 5Kd =1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βp2s−1 (1 − 1/β2ℓ2s−1) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2βp2s (1 − 1/β3ℓ2s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βm−3 (2/β + 1/β3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='− 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='β + 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1≤k≤n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='ik=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='v(1wi1 · · · wik) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1≤k≤n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='ik=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='v(1wi1 · · · wik) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='+ 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βp2s−1+1 v(Xs) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2βp2s+2 (1 − 1/β3ℓ2s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βm−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='=1 − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='β + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βp2s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='3β/2 + 5/2β2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='(1 − 1/β3ℓ2s) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βm−3 (2/β + 5/β2 + 1/β3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βp2s−1 (1 − 1/β2ℓ2s−1) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='βp2s−1+1 v(Xs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='+ 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1≤k≤n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='ik=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='v(1wi1 · · · wik) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='1≤k≤n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='ik=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='v(1wi1 · · · wik) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content='� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' One easily verifies that both 3β/2 + 5/2β2 and 2/β + 5/β2 + 1/β3 equal c := 5 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' We claim that it suffices to show that t � s=1 � 2 βp2s−1 (1 − 1/β2ℓ2s−1) + 5 βp2s−1+1 v(Xs) � (25) + 5 � � � � � 1≤k≤n−1 ik=2 v(1wi1 · · · wik) − � 1≤k≤n−1 ik=0 v(1wi1 · · · wik) � � � � ≤ t � s=1 c βp2s−1 (1 − 1/β2ℓ2s−1), with equality if and only if d ≺ 1(w2w0)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Indeed, suppose the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Then the computation above becomes βv(Ξ(d)) + 5Kd ≤ 1 − 5 β + c t � s=1 � 1 βp2s−1 (1 − 1/β2ℓ2s−1) + 1 βp2s (1 − 1/β3ℓ2s) � + c βm−3 = 1 − 5 β + c t � s=1 (1/βp2s−1 − 1/βp2s + 1/βp2s − 1/βp2s+1) + c βm−3 = 1 − 5 β + c(1/β − 1/βm−3) + c βm−3 = 1 − 5 β + c β = 0 with equality if and only if d ≺ 1(w2w0)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Rearranging, this inequality is equivalent to Kd/βv(Ξ(d)) ≤ −1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' From (18) and the assumption that n(d) = 1, this gives fS(α) = 1 + Kd/βv(Ξ(d)) ≤ 4/5 with equality if and only if d ≺ 1(w2w0)∞, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' 27 It remains to show the claim from (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' The constant c defined above may be rewritten as c = 2+5/(β2+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfmh0R/content/2301.08623v1.pdf'} +page_content=' Subtracting �t s=1(2/βp2s−1)(1 − 1/β2ℓ2s−1) from both sides, dividing by 5 and noting that ik ∈ {0, 2} only when k2s−1 ≤ k < k2s, 1 ≤ s ≤ t, equation (25) becomes t � s=1 � � � � 1 βp2s−1+1 v(Xs) + � k2s−1≤k