MathAudit

Paper metadata

Original submission
November 12, 2007
Claimed MSC
11J54
arXiv categories
math.NT
Comments
8 pages

Abstract

We prove that for any real polynomial f(x)∈R[x]f(x) \in\mathbb{R} [x] the set {α∈R:lim inf⁡n→∞nlog⁡n∣∣αf(n)∣∣>0} \{α\in \mathbb{R}: \liminf_{n\to \infty} n\log n ||αf(n)|| >0\} has positive Hausdorff dimension. Here ∣∣ξ∣∣||ξ|| means the distance from ξξ to the nearest integer. Our result is based on an original method due to Y. Peres and W. Schlag.

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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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements2 reported findingsContains unsupported statements

The construction proves the positive-dimension consequence and the lower bound HD⁡≥γ/(γ+1)\operatorname{HD}\geq\gamma/(\gamma+1), but not the strict inequalities stated in Theorem 1 and its polynomial corollary.

Theorem 1 and CorollaryNot able to verify

The strict Hausdorff-dimension bounds are not established

Pages 1–2 and 7 · Theorem 1, Corollary, and final Eggleston argument · arXiv:0711.1753v1

The theorem claims dimension strictly greater than γ/(γ+1)\gamma/(\gamma+1), and the corollary claims strictly greater than d/(d+1)d/(d+1). The series estimate is valid for every v<γ/(γ+1)v<\gamma/(\gamma+1); at the endpoint, the positive auxiliary exponent cannot be chosen to make the displayed power negative. Eggleston's theorem therefore yields only HD⁡≥γ/(γ+1)\operatorname{HD}\geq\gamma/(\gamma+1). No further argument supplies a strict improvement. The weaker bound is still positive and proves the abstract's central positive-dimension assertion.

Positive-dimension consequenceCorrect

The constructed avoidance set still has positive Hausdorff dimension

Pages 2–7 · Cantor construction and Eggleston estimate · arXiv:0711.1753v1

For every v<γ/(γ+1)v<\gamma/(\gamma+1) the corrected series exponent is negative, so the deletion ratios satisfy Eggleston's convergence condition. Letting vv increase to the endpoint proves HD⁡≥γ/(γ+1)>0\operatorname{HD}\geq\gamma/(\gamma+1)>0. Thus the paper's advertised existence of a positive-dimensional set is supported even though the stronger strict endpoint wording is not.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The endpoint series claim is false and the final proof converts a non-strict dimension estimate into a strict theorem statement.

Endpoint of the dimension argumentIncorrect as written · no repair supplied for strictness

The exponent cannot be negative at the claimed endpoint

Pages 3 and 7 · Equation defining ω\omega and proof of Theorem 1 · arXiv:0711.1753v1

The proof sets ω=(((1+1/γ+ε2)v−1)(γ+1))\omega=(((1+1/\gamma+\varepsilon_2)v-1)(\gamma+1)). At v=γ/(γ+1)v=\gamma/(\gamma+1) every ε2>0\varepsilon_2>0 makes ω>0\omega>0, contrary to the sentence claiming that it can be made negative for v≤γ/(γ+1)v\leq\gamma/(\gamma+1). Restricting to v<γ/(γ+1)v<\gamma/(\gamma+1) repairs the series argument but yields only the non-strict Hausdorff-dimension bound, not the theorem as stated.

Equation (EST)Typo

The product and sum use nn where the running index is required

Page 3 · Equation (EST) · arXiv:0711.1753v1

Inside the product and sum, replace γ/n\gamma/n and the corresponding error term by γ/j\gamma/j and its jj-indexed error. The claimed comparison with (m/n)γ(m/n)^\gamma and the hypothesis on tj+1/tjt_{j+1}/t_j uniquely determine this correction.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0711.1753v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
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