arXiv:math/0210349v1
Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds
math.NT11J8311K60
Abstract
The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The divergence Khintchine--Groshev theorem for nondegenerate manifolds is correct under the paper's monotonicity convention.
Theorem 1.6Correct
Nondegenerate manifolds are of Groshev type for divergence
Page 4 and Section 5 (pages 16--17) · main theorem and final proof · arXiv:math/0210349v1
Local nondegeneracy yields a ball on which sufficiently many integer coefficient vectors have gradients and values in the controlled ranges supplied by the effective upper-bound theorem. Those coefficient vectors define resonant hypersurfaces forming a regular system. The regular-system divergence theorem then gives full measure whenever diverges, and a countable cover of the nondegenerate parameter set proves the global assertion.
Regular-system reductionCorrect
The resonant-set formulation has the required counting and separation
Sections 3--5 (pages 7--17) · Theorems 3.2 and 4.1 and proof of Theorem 1.6 · arXiv:math/0210349v1
The construction selects coefficient vectors in a dyadic height window, obtains a uniform lower derivative, and uses the implicit-function estimate to compare distance to a resonant set with the small linear-form value. The counting lower bound and mutual separation are at the exact scale required by the regular-system theorem, while discarded boundary balls have asymptotically negligible measure.
02Proofs2 reported findingsCorrect
The effective upper bounds, regular-system construction, approximation-by-resonances lemma, and final covering argument are correct and complete.
Section 2Correct and complete
The effective upper-bound inputs cover every derivative regime
Pages 4--6 · Theorems 2.1--2.3 · arXiv:math/0210349v1
The proof partitions integer linear forms according to gradient size and applies the cited quantitative nondivergence estimate only after verifying smoothness, nondegeneracy, and the uniform lower bound. The complementary large-gradient regime is controlled by a direct derivative argument. Their exceptional-set bounds are summable at the chosen scales.
Sections 3--5Correct and complete
The ubiquity argument implies full measure locally and globally
Pages 7--17 · Theorems 3.2 and 4.1 and proof of Theorem 1.6 · arXiv:math/0210349v1
The regular-system theorem converts divergence of the original series to almost-everywhere infinitely many resonant approximations. The distance-to-resonance estimate converts each such approximation back to the paper's linear Diophantine inequality with only fixed constants; monotonicity absorbs those constants. Exhausting the domain by admissible balls completes the proof.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.