arXiv:math/9812088v2
Logarithm laws for flows on homogeneous spaces
math.DSmath.DGmath.NT58F1757S2511J1311J83
Abstract
We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets of a homogeneous space ( a semisimple Lie group, a non-uniform lattice) and a sequence of elements of we prove that for almost all points of the space, one has for infinitely many . The main tool is exponential decay of correlation coefficients of smooth functions on . Besides the aforementioned application to geodesic flows, as a corollary we obtain a new proof of the classical Khinchin-Groshev theorem in simultaneous Diophantine approximation, and settle a related conjecture recently made by M. Skriganov.
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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.
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Detailed mathematical audit
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 , 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.
02Proofs2 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.
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 -regularity by and claims uniform majorants and minorants of cusp indicators with this bound. This is impossible for the claimed minorants as the target measure tends to zero: the positive operator satisfies , while and imply by Cauchy–Schwarz that ; the asserted regularity would instead give at most . The correction arXiv:1706.08570v4 identifies the mistaken Young-inequality step and proves the valid bound 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 ↗03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.