arXiv:2606.28860v1

Maximal Gaps for Dilated Lacunary Integer Sequences

Yuval Peres, Bohan Yang

math.NTmath.DSmath.PR11K3811J8311J7142A55

Paper metadata

Original submission: June 27, 2026
Claimed MSC: 11K38, 11J83, 11J71, 42A55
arXiv categories: math.NT, math.DS, math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 22 pages

Abstract

Let $(a_n)_{n\ge1}\subset\mathbb{N}$ be a lacunary sequence, $a_{n+1}\ge q a_n$ for $q>1$ . For $x\in\mathbb{T}$ , we study the maximal empty circular gap $G_N(x)$ of the finite orbit $\{a_1x,\ldots,a_Nx\}$ . We prove that, for Lebesgue-almost every $x$ , $\frac{1}{2} \le \liminf_{N\to\infty}\frac{NG_N(x)}{\log N} \le \limsup_{N\to\infty}\frac{NG_N(x)}{\log N} \le \frac{q+1}{q-1}\,.$ If, in addition, $a_n\mid a_{n+1}$ for every $n$ , then this can be improved to $\lim_{N\to\infty}\frac{NG_N(x)}{\log N}=1$ for Lebesgue-almost every $x$ .

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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 15, 2026

01Statements2 reported findingsCorrect

The almost-sure order- $\log N/N$ bounds for arbitrary Hadamard-lacunary sequences and the sharp limiting constant for divisibility chains are supported for their full stated ranges.

Theorem 1.1Correct

General lacunary maximal-gap bounds

Pages 2 and 4–15 · Theorem 1.1 and Sections 3–4 · arXiv:2606.28860v1

The moving-target lemma compares joint avoidance with $(1-b)^N$ at an error small enough for a second-moment count of empty shifted cells. Taking cell length $\tau\log N/N$ for every $\tau<1/2$ and interpolating geometric subsequences gives the lower constant $1/2$ . For the upper bound, buffered blocks keep the survivor-set boundary small; the two-point estimate and Paley–Zygmund remove a fixed proportion in each block. The averaged ratio statistic $\Gamma$ controls the off-diagonal term, yielding $\limsup_{N\to\infty}\frac{NG_N(x)}{\log N}\leq1+2\Gamma\leq\frac{q+1}{q-1}$ almost surely.

Theorem 1.2Correct

Divisibility chains have the sharp constant $1$

Pages 2 and 15–21 · Theorem 1.2 and Section 5 · arXiv:2606.28860v1

After the measure-preserving normalization $a_1=1$ , the divisibility chain has independent mixed-radix digits. Intervals with a short multiplicative return form only $O((\log N)^8)$ exceptions; regular intervals obtain the sharp block upper estimate. For the lower estimate, a large separated family of candidate gaps gives zero correlation for short returns and a summable long-return contribution. Suen's inequality and the local lemma produce uniform one- and two-target avoidance asymptotics, and the resulting variance bound is summable on geometric subsequences. Thus the almost-sure liminf and limsup both equal $1$ .

02Proofs3 reported findingsCorrect

The moving-target recursion, empty-cell second moment, buffered-block upper bound, and mixed-radix local-dependence proof are correct and complete. Boundary counts, error exponents, and geometric-sequence interpolations close in the stated parameter ranges.

Lemma 3.2 and Proposition 3.3Correct and complete

Moving-target escape and the lower bound

Pages 5–11 · Lemma 3.2 and Proposition 3.3 · arXiv:2606.28860v1

The remote-past fiber has boundary $O_q(a_kq^{-h})$ , so one-point mixing costs $O_q(q^{-h})$ . The moving-target events are exactly pairwise independent because the corresponding integer endomorphism of the two-torus is surjective. The buffer union bounds therefore give the printed one-step recursion. Summation yields an error $O((\log N)^3/N)$ , and for $\tau<1/2$ this is smaller than the squared main term in the empty-cell second moment. Fubini, Chebyshev, and Borel–Cantelli are then applied on a fixed product probability space.

Lemmas 4.1–4.2 and Proposition 4.3Correct and complete

Buffered-block upper estimate

Pages 12–15 · Section 4 · arXiv:2606.28860v1

Buffers make each prior survivor boundary smaller than the current frequencies by $N^{-10}$ . The first moment is therefore $Ls$ times the survivor measure, while the off-diagonal second moment is bounded by the block average of $a_m/a_n$ , converging uniformly to at most $\Gamma+\varepsilon$ . Iterated Paley–Zygmund contraction gives the stated no-hit exponent. A mesh union bound is summable on geometric subsequences whenever $\tau>\beta+1+2(\Gamma+\varepsilon)$ , and monotonicity of maximal gaps supplies the interpolation to every $N$ .

Lemmas 5.4–5.6 and Proposition 5.7Correct and complete

The divisibility-chain dependence estimates

Pages 17–21 · Sections 5.2–5.3 · arXiv:2606.28860v1

Digit windows have degree $O(\log N)$ . The separated interval family removes all intersections whose integer return ratio is at most $(\log N)^4$ ; the remaining ratios grow geometrically, giving total edge correlation $O(ND p^2+Np/(\log N)^4)$ . The plus and minus grid approximations differ from each target by $O(N^{-20})$ , so the two-sided local-dependence estimate gives relative error $O((\log N)^{-3})$ . The first moment tends to infinity and the normalized variance is summable along $N_m=\lfloor\alpha^m\rfloor$ , completing the sharp lower bound.

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No non-novelty findings.

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Paper: arXiv:2606.28860v1
Authors listed: Yuval Peres, Bohan Yang
Audit date: August 15, 2026

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Maximal Gaps for Dilated Lacunary Integer Sequences

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

General lacunary maximal-gap bounds

Divisibility chains have the sharp constant 111

Moving-target escape and the lower bound

Buffered-block upper estimate

The divisibility-chain dependence estimates

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Authenticate with ORCID, then require an exact arXiv match

Divisibility chains have the sharp constant $1$