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 findingsCorrect
The convergence Khintchine-type theorem for nondegenerate manifolds, including its multiplicative form, is correct under the stated smoothness and summability hypotheses.
Theorem 1.1Correct
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 1.4Correct
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 findingsCorrect
The gradient decomposition, quantitative non-divergence argument, and final Borel-Cantelli summation are correct and complete.
Large-gradient argumentCorrect and complete
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 argumentCorrect and complete
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 non-novelty findings
No non-novelty findings.