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arXiv:1706.08570v4

Distance-like functions and smooth approximations: a correction to "Logarithm laws for flows on homogeneous spaces"

Dmitry Kleinbock, Gregory Margulis

math.DS37A1722F30
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Paper metadata

Original submission
June 26, 2017
Latest revision
August 23, 2017 · version 4
Claimed MSC
37A17, 22F30
arXiv categories
math.DS
Comments
6 pages; minor changes made, misprints corrected

Abstract

One of the propositions in the paper [D. Kleinbock and G.A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), 451-494] related to approximating certain sets by smooth functions, was recently found to be incorrect. Here we correct the mistake.

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Audit summary

Audited against arXiv v4

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 corrected smooth-approximation theorem, the corresponding dynamical Borel–Cantelli estimate, and the recovery of the logarithm-law input from the 1999 paper are correct.

Theorem 1.1Correct

Square-root Sobolev control for cusp approximants

Pages 2–4 · Theorem 1.1 · arXiv:1706.08570v4

Convolution of inner and outer cusp neighborhoods with a fixed smooth kernel gives pointwise minorants and majorants. Young's inequality bounds each derivative in L2L^2 by a constant times the square root of the neighborhood measure, while the distance-like tail comparison gives the required L1L^1 bounds.

Full paper, version 4 ↗
Theorems 1.2 and 1.3Correct

Second-moment Borel–Cantelli and recovery of the original theorem

Pages 3–6 · Theorems 1.2–1.3 · arXiv:1706.08570v4

Exponential mixing bounds a covariance by the product of the square-root Sobolev norms. Splitting pairs according to temporal separation and using exponential divergence makes the total covariance O(EN)O(E_N), so the standard second-moment lemma gives almost-sure normalization. Applying this to the corrected cusp approximants yields the two-sided count estimate and the original Borel–Cantelli conclusion.

02Proofs2 reported findingsCorrect

The convolution estimate, covariance summation, and comparison of smooth approximants with cusp indicators are correct and complete.

Lemma 2.1 and Theorem 1.1Correct and complete

Young's inequality gives the exact repaired regularity

Pages 3–5 · Lemma 2.1 and proof of Theorem 1.1 · arXiv:1706.08570v4

Differentiating the fixed convolution kernel preserves a uniform L1L^1 norm, and Young's inequality converts the indicator's L2L^2 norm to the square root of its measure. Uniform continuity and the distance-like property compare the inner and outer neighborhoods to the same tail probability.

Proofs of Theorems 1.2–1.3Correct and complete

The variance is linear in the expectation

Pages 5–6 · proofs of Theorems 1.2–1.3 · arXiv:1706.08570v4

For separated indices, exponential mixing and the square-root bounds give a summable exponential factor; nearby indices contribute at most their individual expectations. The combined variance is O(EN)O(E_N), and sandwiching the indicators between the two approximants transfers the almost-sure estimate to the cusp events.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1706.08570v4
Authors listed
Dmitry Kleinbock, Gregory Margulis
Audit date
August 19, 2026
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