MathAudit

Paper metadata

Original submission
June 24, 2005
Claimed MSC
11J83, 37A13
arXiv categories
math.NT, math.DS
Comments
LaTeX, 15 pages

Abstract

We prove that singular vectors have measure zero with respect to any friendly measure on Rn\Bbb R^n (e.g. the volume measure on a nondegenerate submanifold). This generalizes special cases considered by Davenport-Schmidt, Baker and Bugeaud. The main tool is quantitative nondivergence estimates for quasi-polynomial flows on homogeneous spaces.

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 19, 2026
01Statements2 reported findingsCorrect

The measure-zero theorem for weighted singular vectors and the existence theorem on analytic manifolds are correct; one auxiliary lemma needs only a harmless small-parameter range correction.

Theorems 1.1 and 1.2Correct

The two main singular-vector results are correct

Pages 2–4 · Theorems 1.1–1.2 · arXiv:math/0506513v1

Friendliness gives the quantitative non-divergence estimate needed to show that weighted singular points have measure zero. For a connected analytic manifold of dimension at least two outside every rational affine subspace, the topological construction produces a totally irrational point with the required weighted singularity.

Full paper, version 1 ↗
Correspondence lemmaMinor formal correction · no status impact

The displayed comparison should be restricted to small epsilon

Pages 5–6 · correspondence lemma · arXiv:math/0506513v1

One implication sets a new parameter equal to a positive power of epsilon and then uses that this new parameter is smaller than epsilon. This is guaranteed for epsilon between zero and one, which is the only regime used to test divergence. Adding that range to the lemma repairs the printed statement and changes no main result.

02Proofs2 reported findingsCorrect

The quantitative non-divergence and topological constructions are correct and complete after the local range correction.

Proof of Theorem 1.1Correct and complete

Friendly-measure estimates give the required null set

Pages 6–10 · proof of Theorem 1.1 · arXiv:math/0506513v1

Federer control and absolute decay imply goodness for the linear functions attached to primitive subgroups. Quantitative non-divergence then gives a summable sequence of exceptional sets, and the unbounded time set is handled with the stated discretization.

Proof of Theorem 1.2Correct and complete after the stated repair

The nested construction enforces singularity and total irrationality

Pages 10–15 · proof of Theorem 1.2 · arXiv:math/0506513v1

The construction alternates prescribed rational relations with avoidance of every previously enumerated rational hyperplane. Analyticity prevents local containment from appearing unexpectedly, and the shrinking neighborhoods yield a point satisfying all required approximation scales.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/0506513v1
Authors listed
Dmitry Kleinbock, Barak Weiss
Audit date
August 19, 2026
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  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
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