arXiv:0904.2795v2
Metric Diophantine approximation for systems of linear forms via dynamics
math.NTmath.DS11J8311J5437A1737A45
Abstract
The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong extremality' of arbitrary finite collection of smooth nondegenerate submanifolds of . The proofs are based on generalized quantitative nondivergence estimates for translates of measures on the space of lattices.
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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 joint strong-extremality theorem for products of nondegenerate maps and its general good/nonplanar-measure formulation are correct.
Theorem 1.2Correct
Joint strong extremality of product maps
Pages 2–4 · Theorem 1.2 · arXiv:0904.2795v2
The product parametrization satisfies the quantitative goodness assumptions factor by factor. Nondegeneracy gives the required exterior-algebra lower bounds, and quantitative nondivergence makes every very-well-multiplicatively-approximable limsup set summable at the relevant scales.
Full paper, version 2 ↗Theorem 2.1Correct
The good/nonplanar metric theorem
Section 2 · Theorem 2.1 · arXiv:0904.2795v2
The theorem's abstract hypotheses are exactly those used in the homogeneous-dynamics reduction: goodness controls small covolumes and nonplanarity rules out an identically small primitive wedge. The conclusion therefore applies to the pushforward product measure with the announced exponent.
02Proofs3 reported findingsCorrect
The dynamical correspondence, quantitative nondivergence argument, and exterior-algebra verification are correct. One acronym in the introductory definition is a uniquely identifiable notation typo.
Proposition 3.1 and Theorem 4.3Correct and complete
Dynamics and nondivergence use the same exponent
Sections 3–4 · Proposition 3.1 and Theorem 4.3 · arXiv:0904.2795v2
The diagonal flow encodes the multiplicative inequalities with the same coordinate weights used in the height function. The quantitative nondivergence estimate is uniform over primitive subgroups, so Borel–Cantelli excludes the claimed exceptional exponent without interchanging quantifiers.
Section 5Correct and complete
The product-map exterior estimates are complete
Section 5 · Proposition 5.4 and proof of Theorem 1.2 · arXiv:0904.2795v2
Expansion of each primitive wedge separates its coordinate polynomials into the product factors. Nondegeneracy supplies a uniform nonzero derivative in every possible wedge component, establishing the required lower bound for all ranks.
Definition of strong extremalityTypo
An undefined acronym replaces the defined approximation class
Page 2 · paragraph defining strong extremality · arXiv:0904.2795v2
The sentence says that almost every matrix is ‘not VVWA,’ while the surrounding text defines very well multiplicatively approximable matrices as VWMA. Replace VVWA by VWMA. The intended correction is unique and the proofs consistently use the multiplicative class.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.