MathAudit

Paper metadata

Original submission
September 23, 2010
Claimed MSC
11K31
arXiv categories
math.NT
Comments
12 pages

Abstract

We prove that there exist arbitrarily small positive real numbers εε such that every integral power (1+ε)n(1+ε)^n is at a distance greater than 2−17ε∣log⁡ε∣−12^{-17}ε|\log ε|^{-1} from the set of rational integers. This is sharp up to the factor 2−17∣log⁡ε∣−12^{-17}|\log ε|^{-1}. We also establish that the set of real numbers α>1α>1 such that the sequence of fractional parts ({αn})n≥1(\{α^n\})_{n\geq 1} is not dense modulo 11 has full Hausdorff dimension.

AI-generated audit

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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements2 reported findingsCorrect

The avoidance bound for powers near one, its translated inhomogeneous form, and the full-dimension interval-confinement theorem are correct.

Theorems 1 and 2Correct

Uniform avoidance at scale ε/∣log⁡ε∣\varepsilon/|\log\varepsilon|

Pages 2-3 and 10 · Theorems 1-2 and conclusion of Section 3 · arXiv:1009.4528v1

The nested dyadic intervals avoid every resonance ∣ekξ−m∣<ψ/2|e^{k\xi}-m|<\psi/2. With ε=eξ−1\varepsilon=e^{\xi}-1 and ψ=2−14ε/∣log⁡ε∣\psi=2^{-14}\varepsilon/|\log\varepsilon| up to the explicit dyadic comparison, the preliminary logarithmic estimate gives the stated lower bound for every positive integer kk. Translation of the resonance centers gives Theorem 2 without changing the counting estimate.

Full paper, version 1 ↗
Theorem 3Correct

Full dimension for prescribed fractional-part intervals

Pages 3 and 10-12 · Theorem 3 and Section 4 · arXiv:1009.4528v1

At level nn, every retained interval has uniformly many children whose nnth powers lie in [an,an+ε][a_n,a_n+\varepsilon]. The separation and contraction estimates permit the standard mass-distribution lower bound, and letting the auxiliary exponent tend to zero makes that lower bound tend to one. Scaling reduces general ξ>0\xi>0 to the written ξ=1\xi=1 construction.

02Proofs2 reported findingsCorrect

The dangerous-set estimate, independence shift, induction, and Cantor construction are correct and complete.

Lemma 1Correct and complete

The independence-shift measure estimate closes

Pages 7-8 · Lemma 1 and equations (3.11)-(3.13) · arXiv:1009.4528v1

The centers log⁡(m)/k\log(m)/k have reciprocal spacing comparable to 1/(km)1/(km), while (3.12) makes the parent interval long enough to contain at least two centers. Summing 1/m1/m between the endpoint indices yields (3.13), and the endpoint remainder is at most the parent length. Together with the dyadic enlargement factor four this gives the claimed 16ψ16\psi bound.

Section 3.5Correct and complete

The dyadic induction preserves half of one child

Pages 8-10 · inductive step and conclusion · arXiv:1009.4528v1

Summing Lemma 1 over the next block of hh indices removes at most 16ψh16\psi h times the parent measure. The numerical choice makes 32ψh≤1/232\psi h\leq1/2, so averaging supplies a dyadic child satisfying both induction conditions. Compact nesting then gives a point outside every dangerous interval.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1009.4528v1
Authors listed
Yann Bugeaud, Nikolay Moshchevitin
Audit date
August 20, 2026
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