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Maximal Gaps for Dilated Lacunary Integer Sequences

T0 review · 0 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read For almost every x, the maximal gap G_N(x) in lacunary orbits satisfies 1/2 ≤ liminf NG_N(x)/log N ≤ limsup NG_N(x)/log N ≤ (q+1)/(q-1).

desk verdict Peres and Yang give explicit constants 1/2 and (q+1)/(q-1) for liminf and limsup of normalized maximal gaps in lacunary sequences, sharpening to limit 1 under divisibility. read the letter →

arxiv 2606.28860 v1 pith:UUTI2KPL submitted 2026-06-27 math.NT math.DSmath.PR

Yuval Peres , Bohan Yang This is my paper · ORCID
classification math.NTmath.DSmath.PR
keywords lacunarysequencesmaximalgapstoralorbitsalmosteverywherecircularuniformdistributioninteger
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that when an integer sequence grows at least geometrically with fixed ratio q > 1, the largest empty arc left by the first N points a1 x through aN x on the circle, scaled by N and divided by log N, has liminf at least 1/2 and limsup at most (q+1)/(q-1) for Lebesgue-almost every starting x. When the sequence is also multiplicative so that each term divides the next, the same normalized gap converges exactly to the value 1 almost everywhere. This quantifies the largest spacing that must appear in these rapidly expanding orbits despite their lacunarity. A reader cares because the constants are explicit and depend only on the minimal growth ratio q, giving sharp control on how non-uniform the points remain.

What carries the argument

The maximal empty circular gap G_N(x) of the finite set {a1 x, …, aN x} mod 1, whose size is tracked after normalization by log N / N.

What would settle it

Numerical computation of NG_N(x)/log N for the sequence a_n = 2^n (where q=2 and the divisibility condition holds) and many random x, checking whether the values approach 1 as N grows to 10^6 or larger.

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Extended reading notes

Core claim

Let (a_n) be a lacunary sequence satisfying a_{n+1} ≥ q a_n for some fixed q > 1. For the maximal empty circular gap G_N(x) among the points {a1 x, …, aN x} on the torus, the paper shows that 1/2 ≤ liminf_{N→∞} N G_N(x)/log N ≤ limsup_{N→∞} N G_N(x)/log N ≤ (q+1)/(q-1) for Lebesgue-almost every x. If in addition a_n divides a_{n+1} for every n, then the limit lim_{N→∞} N G_N(x)/log N equals 1 for Lebesgue-almost every x.

Load-bearing premise

The sequence must obey a fixed geometric lower bound on growth a_{n+1} ≥ q a_n with q > 1, and the statements concern Lebesgue measure on the circle.

Editorial extensions

If this is right

  • Gaps are at most C log N / N with C = (q+1)/(q-1) for all large N, almost everywhere.
  • Gaps of size at least (1/2) log N / N recur infinitely often, almost everywhere.
  • When a_n divides a_{n+1}, the gap size is asymptotically exactly log N / N almost everywhere.
  • The constants depend only on the minimal ratio q and are independent of further details of the sequence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • When q is close to 1 the upper constant grows without bound, allowing arbitrarily large clustering for sequences that barely satisfy the lacunary condition.
  • The sharp limit under divisibility suggests that the extra arithmetic structure removes the fluctuation between lower and upper bounds.
  • The same gap analysis could be tested on non-integer multipliers or on sequences that grow faster than any fixed q.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for a lacunary sequence (a_n) of positive integers satisfying a_{n+1} ≥ q a_n with fixed q > 1, the maximal empty circular gap G_N(x) among the points {a_1 x, …, a_N x} on the torus satisfies 1/2 ≤ liminf_{N→∞} N G_N(x)/log N ≤ limsup_{N→∞} N G_N(x)/log N ≤ (q+1)/(q-1) for Lebesgue-almost every x. Under the additional hypothesis that a_n divides a_{n+1} for every n, the normalized gap converges to 1 almost everywhere.

Significance. The results supply explicit almost-sure bounds on maximal gaps for lacunary orbits that depend only on the lacunarity ratio q, together with a sharpening to an exact limit under divisibility. This contributes to the metric theory of uniform distribution and Diophantine approximation on the circle by furnishing concrete constants rather than existence statements. The derivation is grounded in standard Borel-Cantelli and independence arguments for fractional parts, which are applied without hidden parameters or self-referential quantities.

minor comments (2)
  1. [Introduction] The definition of the maximal gap G_N(x) is introduced early but would benefit from an explicit formula or a short illustrative figure showing the circular arrangement of the points.
  2. [Introduction] A brief outline of the proof strategy (e.g., the role of the lacunarity condition in establishing independence of the events) could be added to the introduction for readers who do not proceed immediately to the technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the clear summary of our results, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained via standard measure theory

full rationale

The paper establishes explicit liminf/limsup bounds for NG_N(x)/log N under the given lacunary hypothesis a_{n+1} ≥ q a_n (q>1) and Lebesgue measure on the torus, using Borel-Cantelli and independence of fractional parts {a_k x}. The constants 1/2 and (q+1)/(q-1) are derived directly from the minimal growth ratio q and the resulting gap statistics; the divisibility case sharpens to the limit 1 by the same techniques. No step reduces a claimed prediction to a fitted input, renames a known result, or relies on a load-bearing self-citation whose justification is internal to the authors' prior unverified work. The derivation chain is independent of the target quantities and externally falsifiable via the stated assumptions.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the standard definition of lacunary sequences and the use of Lebesgue almost-everywhere statements; no free parameters are introduced and no new entities are postulated.

assumptions (1)
  • domain assumption Lebesgue measure on the torus is the natural probability measure for almost-everywhere statements about x
    The result is asserted to hold for Lebesgue-almost every x.

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Cite this review

Pith. "Pith review of Maximal Gaps for Dilated Lacunary Integer Sequences." pith.science (2026). https://pith.science/paper/UUTI2KPL

@misc{pith2026260628860,
  author       = {Pith},
  title        = {Pith review of: Maximal Gaps for Dilated Lacunary Integer Sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UUTI2KPL}},
  note         = {Machine review of arXiv:2606.28860}
}
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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Reference graph

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