Published paper
Abstract
Lemma 3.2 proves the multivariable polynomial-good estimate used for the exterior-power covolume functions in Shi's arithmetic nondivergence argument.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Lemma 3.2 · source p. 8Shows that every degree- polynomial on is -good, verifying the good-function hypothesis in Shi's arithmetic nondivergence branch.
Proof-critical source
Submanifold-genericity of -actions and uniform multiplicative Diophantine approximation
This paper is included only for the following marked statement:
- Lemma 3.2 · source p. 8Polynomial-good estimate used for Shi's exterior-power covolume functions.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:math/0210298v1 · explicit fallback for version of record
Victor Bernik, Dmitry Kleinbock, G. A. Margulis. Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions. International Mathematics Research Notices 2001(9), 453–486 (2001). Reviewed form: arXiv:math/0210298v1.
The dependence graph cites the published article, but the fixed arXiv source is the proof-bearing form previously audited in the corpus. This report reuses that exact source audit and does not claim to audit the version of record.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The convergence Khintchine-type theorem for nondegenerate manifolds, including its multiplicative form, is correct under the stated smoothness and summability hypotheses.
Theorem 1.1No issue found
The convergence theorem has the correct scope
Pages 2–4 · Theorem 1.1 · arXiv:math/0210298v1
The nondegeneracy condition supplies uniform control outside a null exceptional set, and the displayed convergence series is the one required after decomposing integer coefficients into dyadic blocks. The standard and multiplicative conclusions retain the correct almost-everywhere quantifiers.
Full paper, version 1 ↗Theorems 1.3 and 1.4No issue found
The large- and small-gradient estimates support the main theorem
Pages 5–9 · Theorems 1.3–1.4 · arXiv:math/0210298v1
The large-gradient part is controlled by a change-of-variables estimate, while the small-gradient part is reduced to quantitative non-divergence for the associated lattices. Their parameter ranges cover the dyadic decomposition used in Theorem 1.1.
02Proofs2 reported findingsNo proof issues found
The gradient decomposition, quantitative non-divergence argument, and final Borel-Cantelli summation are correct and complete.
Large-gradient argumentNo issue found in the reviewed proof
The local measure estimate is complete
Pages 9–13 · proof of Theorem 1.3 · arXiv:math/0210298v1
A coordinate with uniformly large derivative is selected on each local chart, and Fubini together with one-dimensional monotonicity gives the required measure bound with constants uniform over each dyadic block.
Small-gradient and summation argumentNo issue found in the reviewed proof
Quantitative non-divergence gives a summable exceptional set
Pages 13–27 · proof of Theorem 1.4 and completion of Theorem 1.1 · arXiv:math/0210298v1
The exterior-power lower bounds required by quantitative non-divergence follow from nondegeneracy, and the resulting estimates remain summable after the multiplicative dyadic partition. Borel-Cantelli then yields exactly the claimed null set.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.