MathAudit

Paper metadata

Original submission
December 21, 1999
Claimed MSC
37B10, 37D20
arXiv categories
math.DS
Comments
Latex, 22 pages
Journal reference
Israel J. Math. 122 (2001), 1--27.

Abstract

Let T:X↦XT: X\mapsto X be a deterministic dynamical system preserving a probability measure μμ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsets An⊂XA_n\subset X and μμ-almost every point x∈Xx\in X the inclusion Tnx∈AnT^nx\in A_n holds for infinitely many nn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.

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 sufficient dynamical Borel-Cantelli criteria, their Gibbs-measure applications, and the counterexamples separating the stated hypotheses are correct as stated.

Theorems 2.1 and 2.2Correct

The abstract Borel-Cantelli criteria are correct

Pages 4–8 · Theorems 2.1–2.2 · arXiv:math/9912178v1

The summability and overlap hypotheses give the announced strong Borel-Cantelli conclusion through the second-moment estimate, while the weaker hypothesis gives the ordinary Borel-Cantelli property. The quantifiers over the target sequence and the normalization by the cumulative measure agree with the conclusions.

Full paper, version 1 ↗
Cylinder and rectangle applicationsCorrect

The Gibbs applications and limitations match the abstract criteria

Pages 8–23 · application theorems and counterexamples · arXiv:math/9912178v1

Exponential mixing supplies the required correlation bounds for the nested cylinder and aligned rectangle families. The constructed counterexamples remove precisely the geometric alignment or nesting control used in those estimates and therefore do not conflict with the positive results.

02Proofs2 reported findingsCorrect

The variance estimates, Gibbs distortion bounds, and counterexample constructions are correct and complete.

Proofs of Theorems 2.1 and 2.2Correct and complete

The correlation sums close with the stated normalizations

Pages 5–8 · proofs of Theorems 2.1–2.2 · arXiv:math/9912178v1

The diagonal and off-diagonal contributions are separated correctly, the assumed overlap bounds control the variance, and the standard subsequence interpolation produces the full strong law without changing the target sequence.

Sections 3–4Correct and complete

The symbolic and geometric reductions are complete

Pages 8–23 · Gibbs applications and examples · arXiv:math/9912178v1

Bounded distortion converts cylinder intersections into the abstract correlation estimates, and the rectangle argument tracks stable and unstable coordinates separately. The negative examples explicitly verify divergence of the measure sum while forcing failure of the asserted recurrence conclusion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/9912178v1
Authors listed
Nikolai Chernov, Dmitry Kleinbock
Audit date
August 19, 2026
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