MathAudit

Publication metadata

Journal citation
Inventiones Mathematicae 138(3), 451–494 (1999)
Publication date
1999
arXiv counterpart
arXiv:math/9812088v2

Abstract

The finite-cover lemma used in this graph shows that the relevant spectral property is preserved under passage between finite-index and commensurable lattices.

Role in dependence graphs

Proof-critical source

Bounded trajectories of quasi-rays

This paper is included only for the following marked statement:

  • Lemma 3.1 · printed p. 460; proof Appendix Proposition A.0 and Lemma A.3, pp. 480–485Transfers spectral gap across finite covers and commensurable lattices.

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Audited against an explicitly disclosed arXiv fallback

Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.

Exact reviewed source

arXiv:math/9812088v2 · explicit fallback for inaccessible version of record

Dmitry Kleinbock, Grigory Margulis. Logarithm laws for flows on homogeneous spaces. Inventiones Mathematicae 138 (1999), no. 3, 451–494. Reviewed form: arXiv:math/9812088v2.

Springer's version of record is subscription-controlled. The author-hosted prepublication PDF is not byte-identical to arXiv v2 and is not identified as the version of record, so neither accessible file can safely be represented as the VOR. This report therefore reuses the existing audit of the fixed arXiv v2 source and does not claim to audit the VOR.

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Generated August 23, 2026
01Statements2 reported findingsCorrect

The logarithm laws and Borel–Cantelli conclusions are correct, but this conclusion relies on the verified replacement argument published in arXiv:1706.08570v4: the proof printed in the reviewed version uses a false smooth-approximation lemma.

Theorems 1.4, 1.7, and 1.9Correct

Logarithm laws for homogeneous and locally symmetric spaces

Sections 1 and 4–6 · arXiv:math/9812088v2

The later correction proves the smooth approximation bound with ∥h∥2,ℓ≤C∥h∥1\lVert h\rVert_{2,\ell}\leq C\sqrt{\lVert h\rVert_1}, establishes a compatible second-moment Borel–Cantelli theorem under exponential mixing, and explicitly recovers the original Theorem 4.3. The geometric reduction in Sections 5–6 then gives the stated logarithm laws without changing their hypotheses or constants.

Correction, version 4 ↗
Theorems 8.1–8.3Correct

Metric Diophantine applications

Section 8 · Theorems 8.1–8.3 · arXiv:math/9812088v2

The lattice-point identities and volume integrals in Sections 7–8 convert the Diophantine inequalities into the same cusp events governed by the repaired Borel–Cantelli theorem. Thus the zero-one and logarithm conclusions follow from a verified repair even though the original analytic input is invalid as printed.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The proof is incorrect as written. Lemma 4.2 claims a Sobolev bound linear in the target measure, and the proof of Theorem 4.3 applies Proposition 4.1 using that false input. The authors' later correction supplies the valid square-root bound and a new second-moment argument that recovers the central results. The focal change-of-variables Lemma 8.3 is self-contained and unaffected.

Lemma 4.2Incorrect as written · verified repair

The asserted smooth-approximation regularity is false

Pages 11–12 · Lemma 4.2 · arXiv:math/9812088v2

The paper defines (C,ℓ)(C,\ell)-regularity by ∥Υℓh∥2≤Cμ(h)\lVert\Upsilon^\ell h\rVert_2\leq C\mu(h) and claims uniform majorants and minorants of cusp indicators with this bound. This is impossible for the claimed minorants as the target measure aa tends to zero: the positive operator Υ=1−∑Xi2\Upsilon=1-\sum X_i^2 satisfies ∥Υℓh′∥2≥∥h′∥2\lVert\Upsilon^\ell h'\rVert_2\geq\lVert h'\rVert_2, while 0≤h′≤1A0\leq h'\leq1_A and ∥h′∥1≥ca\lVert h'\rVert_1\geq ca imply by Cauchy–Schwarz that ∥h′∥2≥∥h′∥1/a≥ca\lVert h'\rVert_2\geq\lVert h'\rVert_1/\sqrt a\geq c\sqrt a; the asserted regularity would instead give at most C∥h′∥1≤CaC\lVert h'\rVert_1\leq Ca. The correction arXiv:1706.08570v4 identifies the mistaken Young-inequality step and proves the valid bound ∥h∥2,ℓ≤C∥h∥1\lVert h\rVert_{2,\ell}\leq C\sqrt{\lVert h\rVert_1} as Theorem 1.1. Repair classification: Verified repair.

Correction, version 4 ↗
Application of Proposition 4.1 in Theorem 4.3Incorrect as written · verified repair

The printed Borel–Cantelli proof applies a valid proposition through a false input

Pages 11–13 · Proposition 4.1, Lemma 4.2, and Theorem 4.3 · arXiv:math/9812088v2

Proposition 4.1 is correct under its stated linear-regularity hypothesis, but Lemma 4.2 does not provide functions satisfying that hypothesis. With only the valid square-root estimates, the correlation sum displayed in Proposition 4.1's proof cannot be reused in the proof of Theorem 4.3. Repair classification: Verified repair. Theorems 1.2–1.3 of arXiv:1706.08570v4 prove a new variance estimate under the same exponential-mixing and exponentially-divergent-sequence hypotheses and explicitly recover Theorem 4.3; all downstream central results can therefore use that replacement.

Correction, version 4 ↗
Lemma 8.3Correct and complete

The focal approximation-function change of variables is self-contained

Pages 23–24 · Lemma 8.3 · arXiv:math/9812088v2

Writing P(λ)=−log⁡ψ(eλ)P(\lambda)=-\log\psi(e^\lambda), the proof intersects the nondecreasing graph L=P(λ)L=P(\lambda) with the decreasing affine line determined by tt, then sets r(t)=(L(t)−λ(t))/(m+n)r(t)=(L(t)-\lambda(t))/(m+n). This gives the required monotonicity and the identity ψ(et−nr(t))=e−t−mr(t)\psi(e^{t-nr(t)})=e^{-t-mr(t)}. The inverse construction uses the strictly increasing λ(t)\lambda(t), and the Stieltjes-integral comparison proves convergence equivalence. No external result is used, so the marked BFK dependency terminates at this node.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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