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arXiv:0709.3419v2

Density modulo 1 of sublacunary sequences: application of Peres-Schlag's arguments

Nikolai G. Moshchevitin

math.NT11B8311J25
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Paper metadata

Original submission
September 21, 2007
Latest revision
October 20, 2007 · version 2
Claimed MSC
11B83, 11J25
arXiv categories
math.NT
Comments
9 pages, minor correction in Section 6B

Abstract

Let the sequence {tn}n=1∞\{t_n\}_{n=1}^{\infty} of reals satisfy the condition tn+1tn≥1+γnβ,0≤β<1,γ>0. \frac{t_{n+1}}{t_n} \ge 1+ \fracγ{n^β},0\le β<1, γ>0. Then the set {α∈[0,1]:∃ϰ>0∀n∈N∣∣tnα∣∣>ϰnβlog⁡(n+1)} \{α\in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||t_n α|| > \frac{\varkappa}{n^β\log (n+1)} \} is uncountable. Moreover its Hausdorff dimension is equal to 1. Consider the set of naturals of the form 2n3m2^n3^m and let the sequence s1=1,s2=2,s3=3,s4=4,s5=6,s6=8,... s_1=1, s_2=2, s_3=3, s_4=4, s_5=6, s_6 = 8,... performs this set as an increasing sequence. Then the set {α∈[0,1]:∃ϰ>0∀n∈N∣∣snα∣∣>ϰnlog⁡(n+1)} \{α\in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||s_n α|| > \frac{\varkappa}{\sqrt{n}\log (n+1)} \} also has Hausdorff dimension equal to 1. The results obtained use an original approach due to Y. Peres and W. Schlag.

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Audit summary

Audited against arXiv v2

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 20, 2026
01Statements2 reported findingsContains unsupported statements

The nonemptiness conclusions are recoverable after a constant correction, but the full-dimension conclusion under only the lower sublacunary growth hypothesis is asserted without the construction needed to verify Theorem 3's series condition.

Full-dimension sublacunary applicationNot able to verify

Removal of the upper growth hypothesis is not proved

Pages 3, 6, and 8 · Theorem 3, its sketched proof, and Example A · arXiv:0709.3419v2

The displayed verification of the Hausdorff-dimension series uses the additional upper bound tn+1/tn≤1+γ2/nβt_{n+1}/t_n\leq1+\gamma_2/n^\beta. The paper then says that a different h(n)h(n) removes this assumption but gives neither that function nor checks Theorem 3's monotonicity, separation, summability, and initial conditions. This is a substantive missing construction, so the dimension-one claim under the lower bound alone is not verified by the supplied argument.

Nonemptiness applicationCorrect

The existential avoidance result survives the dyadic correction

Pages 2–5 and 7–8 · Theorems 1–2 and Example A · arXiv:0709.3419v2

Replacing the false factor-two scale estimate by the factor-four estimate recorded below and shrinking the freely chosen avoidance constant by a fixed factor restores the induction. Since the application asserts existence of some positive ϰ\varkappa, this constant change does not alter its statement.

02Proofs2 reported findingsContains incorrect or incomplete proofs

Lemma 1 uses a false dyadic inequality, and the proof of Theorem 3 is only a template that does not verify its hypotheses in the strongest application.

Lemma 1Incorrect as written · verified repair for the existential applications

The dyadic floor inequality is reversed at the endpoint scale

Page 4 · Equations (1)–(2) in the proof of Lemma 1 · arXiv:0709.3419v2

From ln=⌊log⁡2(tn/(2δ(n)))⌋l_n=\lfloor\log_2(t_n/(2\delta(n)))\rfloor one obtains tn/2ln<4δ(n)t_n/2^{l_n}<4\delta(n), not the printed upper bound 2δ(n)2\delta(n). Lemma 1's coefficient must therefore be enlarged from 4δ(n)4\delta(n) to 8δ(n)8\delta(n). Replacing the denominators 44 in the later summability hypotheses by 88, or equivalently halving the freely selected function δ(n)\delta(n) in the applications, restores Theorems 1–2 and their qualitative avoidance conclusions. It does not by itself supply the missing full-dimension construction.

Example A notationTypo

Two displayed parameter symbols have unique corrections

Pages 7–8 · verification of Example A · arXiv:0709.3419v2

Replace the impossible condition γ1<γ1\gamma_1<\gamma_1 by γ1<γ\gamma_1<\gamma, and replace the displayed derivative equality by a lower bound: the omitted positive term makes the derivative strictly larger than the final expression. Both corrections are fixed by the immediately preceding definitions and preserve the monotonicity conclusion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0709.3419v2
Authors listed
Nikolai G. Moshchevitin
Audit date
August 20, 2026
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