MathAudit

Paper metadata

Original submission
August 15, 2007
Latest revision
August 15, 2007 · version 2
Claimed MSC
11B05
arXiv categories
math.NT
Comments
9 pages

Abstract

We present a proof of a multidimensional version of Peres-Schlag's theorem on Diophantine approximations with lacunary sequences.

AI-generated audit

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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements3 reported findingsCorrect

The multidimensional lacunary avoidance theorems and the complex-power corollary follow from the dyadic induction after the uniquely determined displayed corrections below.

Theorem 1Correct

Coordinatewise lacunary avoidance

Pages 2 and 4–7 · Theorem 1 and Lemma 1 · arXiv:0708.2087v2

The principal counting lemma bounds the new dangerous boxes relative to the surviving set. With the corrected displayed coefficients, the Lovász-style measure induction retains positive measure at each scale, and compactness supplies a point whose every coordinate avoids the prescribed lacunary sequence by the stated uniform amount.

Theorem 2Correct

Orthogonal-matrix lacunary avoidance

Pages 3 and 7–8 · Theorem 2 and Lemma 2 · arXiv:0708.2087v2

Orthogonality preserves Euclidean distances and volumes, so each rotated dangerous neighborhood has the same covering bound as in Lemma 1. Lemma 2 repeats the same finite-dependency measure induction for the matrix sequence and yields the simultaneous coordinate separation asserted in Theorem 2.

CorollaryTypo

Complex powers use the standard rotation matrix

Pages 3–4 · complex-number corollary · arXiv:0708.2087v2

Replace the printed reflection matrix by ∣θ∣−1(a−bba)|\theta|^{-1}\begin{pmatrix}a&-b\\ b&a\end{pmatrix}, or equivalently take its jj-th powers as the orthogonal sequence. This is exactly the real matrix for multiplication by θ/∣θ∣\theta/|\theta| and makes Theorem 2 apply to θj\theta^j without changing the corollary.

02Proofs1 reported findingCorrect

The induction and its rotated analogue are complete. Two coefficient displays are typographical and are fixed by the same calculation and by the already stated final constant.

Lemma 1 coefficientsTypo

The covering factors are dropped and one exponent is misprinted

Page 5 · Equations (1) and (3) in the proof of Lemma 1 · arXiv:0708.2087v2

The first line temporarily drops the factor 22d2^{2d} and Equation (3) prints 2d+1δd2^{d+1}\delta^d instead of 22dδd2^{2d}\delta^d. Restoring these factors makes the two contributions at most 24dδd2^{4d}\delta^d each, exactly giving the declared δ1=24d+1δd\delta_1=2^{4d+1}\delta^d. No later bound changes.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0708.2087v2
Authors listed
Nikolai G. Moshchevitin
Audit date
August 20, 2026
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