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arXiv:1101.5032v2

On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations

Nikolay Moshchevitin

math.NT11J20
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Paper metadata

Original submission
January 26, 2011
Latest revision
February 11, 2011 · version 2
Claimed MSC
11J20
arXiv categories
math.NT
Comments
18 pages

Abstract

We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine (1926) and to an original method invented by Y. Peres and W. Schlag (2001).

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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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements2 reported findingsCorrect

The two inhomogeneous avoidance theorems and their stated parameter specializations are correct; the preliminary notation contains two harmless symbol slips.

Theorems 1 and 2Correct

Abstract inhomogeneous product-avoidance criteria

Pages 2-3 and 15-18 · Theorems 1-2 and common Peres-Schlag argument · arXiv:1101.5032v2

The hypotheses bound the aggregate lengths of dangerous intervals on every multiplicative block. Lemmas 1-4 convert the Diophantine lower bounds on α\alpha into those block sums, and Lemma 5 inductively retains positive measure. A point in the nested complements gives exactly inequalities (13) and (16).

Full paper, version 2 ↗
Equations (1) and (3)Typos · no status impact

Two preliminary function names are typographical errors

Page 1 · equations (1)-(3) and preceding definitions · arXiv:1101.5032v2

In (1), the undefined ω\omega is uniquely ω1\omega_1, as confirmed by the equivalent formulation (2) using ω1∗\omega_1^*. In the list before (3), ϕ2\phi_2 is printed twice; the normalization in (3) and every subsequent use show that the first occurrence must be ϕ1\phi_1.

02Proofs3 reported findingsCorrect

The lattice-point counts and the measure-retention induction verify both main theorems; two local labels are harmless typos.

Lemmas 1-4Correct and complete

Counting bounds imply the required block estimates

Pages 12-14 · Lemmas 1-4 · arXiv:1101.5032v2

The integer points in each strip either span a positive-area polygon, where the area bound applies, or lie on one affine lattice line, where condition (2) bounds their step count. Summation over dyadic error levels yields TA,ε[1]≤2−6T^{[1]}_{A,\varepsilon}\leq2^{-6} and the analogous TA,ε[2]T^{[2]}_{A,\varepsilon} estimate with the printed safety factors.

Lemma 2 and Section 8 notationTypos · no status impact

Two set labels name the wrong already-defined object

Pages 13 and 15 · proof of Lemma 2 and definition after (34) · arXiv:1101.5032v2

The collinear case of Lemma 2 must conclude with card⁡Aν(t)\operatorname{card}A_\nu(t), not card⁡Aν,μ\operatorname{card}A_{\nu,\mu}. After (34), the covered union is E[j](x)E^{[j]}(x), not the undefined Eα(x)E_\alpha(x). Both corrections are forced by the immediately preceding definitions and do not alter an estimate.

Lemma 5Correct and complete

The common Peres-Schlag induction retains positive measure

Pages 15-18 · Lemma 5 · arXiv:1101.5032v2

Dyadic enlargement contributes only a fixed factor, and Lemmas 3-4 bound the sum of new dangerous lengths by a strict fraction of the preceding survivor. The scale condition guarantees the needed independence gap, so induction gives nonempty nested closed sets for either family of dangerous intervals.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1101.5032v2
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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