MathAudit

Paper metadata

Original submission
June 24, 2005
Claimed MSC
11J83, 37A13
arXiv categories
math.NT, math.DS
Comments
AMSTeX, 12 pages

Abstract

We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on Rn\Bbb R^n. The correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces is discussed in an elementary way, and several recent results obtained by means of this correspondence are surveyed.

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 dynamical formulation of Diophantine exponents of measures and the stated extremality consequences are correct; one notation mismatch is a harmless typo.

Theorems 3.1 and 3.2Correct

Quantitative non-divergence gives the announced extremality criterion

Pages 6–10 · Theorems 3.1–3.2 · arXiv:math/0506512v1

Goodness controls the small-value sets of the relevant exterior-product functions, while nonplanarity supplies the required uniform lower bounds. Applied to the lattice trajectory associated with the map, these hypotheses imply extremality of the pushforward measure.

Full paper, version 1 ↗
Definition of a good functionTypo · no status impact

The two spellings of the epsilon parameter should agree

Page 6 · definition preceding Theorem 3.1 · arXiv:math/0506512v1

The quantifier introduces one spelling of the epsilon parameter while the displayed numerator uses another spelling. They denote the same positive real parameter, and replacing either occurrence by the other uniquely repairs the notation without changing the definition or any argument.

02Proofs2 reported findingsCorrect

The lattice correspondence and quantitative non-divergence applications are correct and complete at the stated expository level.

Dynamical correspondenceCorrect and complete

The exponent inequalities are translated with the correct time scaling

Pages 3–6 · dynamical reformulation · arXiv:math/0506512v1

The diagonal weights balance the linear-form error against coefficient size, and Mahler's criterion converts unusually good approximation into the asserted cusp excursions. The limiting exponent is unchanged by the reparameterization.

Application of Theorem 3.1Correct and complete after the notation correction

Goodness and nonplanarity verify every non-divergence hypothesis

Pages 8–12 · proof and consequences of Theorem 3.2 · arXiv:math/0506512v1

All primitive subgroups are represented by exterior products, the goodness estimates are uniform on a smaller ball, and nonplanarity prevents an exterior product from vanishing identically. The notation typo does not enter any estimate.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/0506512v1
Authors listed
Dmitry Kleinbock
Audit date
August 19, 2026
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  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
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