arXiv:2608.01453v1

Visiting time statistics

Maxim Kirsebom, Philipp Kunde, Tomas Persson

math.DS37A5037B2037E05

Paper metadata

Original submission: August 2, 2026
Claimed MSC: 37A50, 37B20, 37E05
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 26 pages, 1 figure

Abstract

Many mixing dynamical systems $(X,T,μ)$ are known to satisfy the hitting time statistics result $\lim_{r \to 0} μ\{\, x : τ_{B(y,r)} (x) > t/μ(B(y,r))\,\} = e^{-t},$ for $μ$ -almost every $y$ , where $τ_{B(y,r)} (x)$ is the first hitting time of $x$ to the ball $B(y,r)$ . Taking a different point of view, we fix $x$ and consider $τ_{B(y,r)} (x)$ as a function of $y$ . We call this the visiting time of $y$ from $x$ , i.e. the time it takes for $y$ to get a visit from $x$ within a neighbourhood of radius $r$ . We prove that $\lim_{r \to 0} μ\{\, y : τ_{B(y,r)} (x) > t/μ(B(y,r)) \,\} = e^{-t},$ for $μ$ -almost every $x$ . As a byproduct we obtain a new method of proof for hitting time statistics.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsCorrect

The visiting-time exponential law for finite-branch piecewise expanding interval maps, the fixed-target hitting-time law under logarithmic short-return exclusion, and the associated Poisson law are correct under the stated mixing and regularity hypotheses.

Theorems 1–2Correct

Almost-sure visiting-time statistics

Pages 3–4 and 4–21 · Theorems 1–2 and their proofs · arXiv:2608.01453v1

For radii normalized by $\mu(B(y,r_n(y,t)))=t/n$ , the inclusion–exclusion expansion is reduced to tuples with logarithmically separated times by the almost-everywhere short-return input. Exponential mixing controls the separated correlations, the second-moment estimate is summable along geometric subsequences, and the monotonicity interpolation gives convergence for every $t\geq0$ on one full-measure set of starting points. The final radius interpolation uses the two neighboring normalized radii and Egorov's theorem to pass from $r_n(y,t)$ to an arbitrary sequence $r\downarrow0$ , preserving the quantifier 'for almost every $x$ , for all $t$ '.

Full paper, version 1 ↗

Theorems 3–4Correct

Hitting-time and Poisson statistics

Page 4 and pages 21–26 · Theorems 3–4 and Sections 5–6 · arXiv:2608.01453v1

The assumed exclusion $B(y,r)\cap T^kB(y,r)=\varnothing$ for $k\leq-c_1\log r$ removes every close-time cluster in the inclusion–exclusion sums. The same separated-tuple estimates then give the exponential hitting law. For the Poisson refinement, the probability-generating polynomial has coefficients equal to the hit-count probabilities; the coefficient bounds are uniform enough to pass from convergence to $e^{-tz}$ to the coefficient limit $t^pe^{-t}/p!$ for every fixed $p$ .

02Proofs3 reported findingsCorrect

The central proof chains are correct and complete. The combinatorial estimates are uniform throughout the truncation range $m\leq(\log n)^\alpha$ , and the choices of the three logarithmic gap constants leave a positive decay exponent in the final second-moment bound.

Propositions 1–3Correct and complete

Second-moment proof for visiting times

Pages 8–19 · Propositions 1–3 and Lemmas 2–5 · arXiv:2608.01453v1

Bonferroni truncation brackets the visiting indicator by odd and even sums. The proportion of tuples with intermediate gaps is $O((\log n)^{p(1+2\alpha)}/n^p)$ , while exponential mixing gives errors $n^{-\tau c_i}$ at the designated gap scales. In the paired second moment, gaps shorter than $c_0\log n$ necessarily join one time from each tuple and cannot be consecutive; this justifies grouping those two indicators and bounds the resulting variation growth. With $c_0<c_1/4$ and the stated small choice of $c_0$ , the remaining exponent $s=\tau c_0-\epsilon$ is positive, yielding $\Lambda_n=O(n^{-s})$ .

Proposition 1 interpolationCorrect and complete

Geometric subsequences extend to all sample sizes and all intensities

Pages 8–9 · proof of Proposition 1 · arXiv:2608.01453v1

Summability of $\Lambda_{\lceil a^k\rceil}$ and Borel–Cantelli give almost-sure convergence on every rational geometric subsequence and rational $t>0$ . The normalized balls are nested as their masses vary, so the neighboring subsequences squeeze arbitrary $n$ between intensities $t/a^2$ and $a^2t$ . Letting rational $a\downarrow1$ and then using monotonicity in $t$ establishes the simultaneous conclusion without an uncountable intersection of exceptional sets.

Section 4Correct and complete

Passage from mass-normalized radii to ordinary radii

Pages 19–21 · proof of Theorem 1 · arXiv:2608.01453v1

The two index functions selecting neighboring radii tend to infinity pointwise. Egorov's theorem makes this uniform outside sets of arbitrarily small measure, and the factors $n/(n+1)$ are then uniformly close to one. The upper and lower inclusions use the correct direction of both ball inclusion and hitting-time monotonicity, so the normalized tail probability is squeezed to $e^{-t}$ for every sequence tending to zero.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.01453v1
Authors listed: Maxim Kirsebom, Philipp Kunde, Tomas Persson
Audit date: August 18, 2026

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Visiting time statistics

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Almost-sure visiting-time statistics

Hitting-time and Poisson statistics

Second-moment proof for visiting times

Geometric subsequences extend to all sample sizes and all intensities

Passage from mass-normalized radii to ordinary radii

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