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arXiv:1302.6560v2

Non-planarity and metric Diophantine approximation for systems of linear forms

Victor Beresnevich, Dmitry Kleinbock, Gregory Margulis

math.NTmath.DS
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Paper metadata

Original submission
February 26, 2013
Latest revision
October 18, 2013 · version 2
Claimed MSC
arXiv categories
math.NT, math.DS
Comments
27 pages; one of the theorems from the previous version has been removed

Abstract

In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of m×nm\times n matrices over R\Bbb R is introduced and studied. This notion generalises the one of non-planarity in Rn\Bbb R^n and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.

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Audited against arXiv v2

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.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

The weak-nonplanarity criterion for strong extremality of systems of linear forms and its analytic-manifold consequences are correct.

Theorem 2.3Correct

Goodness plus weak nonplanarity implies strong extremality

Pages 6–7 and 13–20 · Main Theorem 2.3 · arXiv:1302.6560v2

Weak nonplanarity prevents the determinant functions governing rational obstructions from vanishing identically, while goodness controls their small-value sets. The homogeneous-dynamics criterion then rules out very well multiplicatively approximable matrices almost everywhere.

Full paper, version 2 ↗
Corollary 2.4 and product resultsCorrect

The analytic and product applications satisfy the criterion

Pages 7 and 20–27 · analytic and product applications · arXiv:1302.6560v2

Analyticity upgrades failure on a positive-measure set to an identically vanishing determinant obstruction, exactly excluded by weak nonplanarity. The block-product construction preserves the same determinant condition and therefore the claimed extremality.

02Proofs2 reported findingsCorrect

The exterior-algebra non-divergence proof and the verification of weak nonplanarity are correct and complete.

Theorem 4.3Correct and complete

Quantitative non-divergence is applied with every primitive subgroup controlled

Pages 13–18 · Theorem 4.3 and proof · arXiv:1302.6560v2

The exterior coordinates split into the required minors, goodness supplies the uniform small-value estimate, and weak nonplanarity gives a positive supremum on each ball. The resulting cusp estimate is uniform over the relevant diagonal parameters.

Sections 5–6Correct and complete

The analytic and inheritance arguments close

Pages 18–27 · products, transposes, and submanifolds · arXiv:1302.6560v2

Transpose invariance is checked directly at the determinant level, product maps are handled blockwise, and analytic continuation supplies the needed zero-set dichotomy. These reductions cover all cases invoked by the main corollaries.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1302.6560v2
Authors listed
Victor Beresnevich, Dmitry Kleinbock, Gregory Margulis
Audit date
August 19, 2026
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