arXiv:math/0210298v1
Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions
math.NTmath.DS11J83
Abstract
An analogue of the convergence part of the Khintchine-Groshev theorem, as well as its multiplicative version, is proved for nondegenerate smooth submanifolds in . The proof combines methods from metric number theory with a new approach involving the geometry of lattices in Euclidean spaces.
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Detailed mathematical audit
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.