MathAudit

Paper metadata

Original submission
January 31, 2024
Latest revision
March 1, 2024 · version 2
Claimed MSC
arXiv categories
math.NT, math.DS

Abstract

For an m by n real matrix A, we investigate the set of badly approximable targets for A as a subset of the m-torus. It is well known that this set is large in the sense that it is dense and has full Hausdorff dimension. We investigate the relationship between its measure and Diophantine properties of A. On the one hand, we give the first examples of a non-singular matrix A such that the set of badly approximable targets has full measure with respect to some non-trivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on A that slightly strengthens non-singularity, and show that under the assumption that A satisfies this condition, the set of badly approximable targets is a null-set with respect to any non-trivial algebraic measure on the torus. For this we use naive homogeneous dynamics, harmonic analysis, and a novel concept we refer to as mixing convergence of measures.

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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 counterexamples with non-singular matrices, the nullity theorem under the full accumulation-chain hypothesis, and the grid value-set formulation are correct. The local coefficient, ambient-space, and coordinate slips listed below are typographical.

Theorems 1.7, 1.14, and 2.11Correct

The positive- and zero-measure phenomena are correctly separated

Pages 3–13 and 20–38 · main theorems and their proofs · arXiv:2402.00196v2

The parametric-geometry construction gives a non-singular vector whose badly approximable targets contain the announced subtorus coset. In the opposite direction, the accumulation chain yields mixing convergence along the successive lattices, forcing almost every algebraic translate to have dense values and hence to lie outside the badly approximable set.

Full paper, version 2 ↗
Definitions and AppendixTypos · no status impact

The coefficient bounds and ambient coordinates contain local typos

Pages 13, 29, 33, and 38–39 · proof setup, Definitions 7.1 and 8.1, and Proposition 9.2 · arXiv:2402.00196v2

Definition 7.1 must require 0<max⁡i∣ai∣≤t0<\max_i|a_i|\leq t, not that every coefficient is nonzero; Definition 8.1 similarly needs 0<∥q∥≤t0<\lVert q\rVert\leq t. In the proof on page 13 the displayed spaces of qiq_i and pip_i are interchanged, and page 38 again places pp in the wrong ambient integer space. Finally, the maximum in Proposition 9.2 repeats ∣a1∣|a_1| where its second entry must be ∣a2∣|a_2|. Each repair follows directly from the defining matrix dimensions and the adjacent formula.

02Proofs2 reported findingsCorrect

The homogeneous-dynamics inheritance argument, harmonic-analysis step, parametric-geometry construction, and appendices are correct and complete after the local dimension corrections.

Theorem 2.11Correct and complete

The accumulation sequence forces dense values for algebraic translates

Pages 12–13 and 27–29 · Theorem 2.11 and its proof · arXiv:2402.00196v2

Iterating the measure-dimension jump along an accumulation sequence of length dd produces Haar measure on the terminal torus. The coset lemma then places a codimension-one subtorus coset in almost every orbit closure, and the product-value proposition turns that inclusion into the stated dense-value conclusion. The deduction uses only the hypotheses printed in Theorem 2.11.

Sections 2–8Correct and complete after the stated repairs

The accumulation-chain and construction proof chains close

Pages 8–38 · grid reformulation, mixing convergence, and examples · arXiv:2402.00196v2

Each inherited limit measure has the announced rational invariance, coprimality of mm and nn forces the required growth of invariant directions, and the terminal torus gives dense product values. The template construction separately verifies non-singularity and containment of the codimension-two coset.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2402.00196v2
Authors listed
Nikolay Moshchevitin, Anurag Rao, Uri Shapira
Audit date
August 20, 2026
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