MathAudit

Paper metadata

Original submission
October 23, 2002
Claimed MSC
11J83
arXiv categories
math.NT, math.DS
Comments
Preprint, 12 pages
Journal reference
Algebraic groups and arithmetic, 539-553, TIFR, Mumbai, 2004.

Abstract

We show a large class of analytic submanifolds of C^n to be strongly extremal. This generalizes V. Sprindzhuk's solution of the complex case of Mahler's Problem, and settles complex analogues of conjectures made in the 1970s by Baker and Sprindzhuk. The proof is based on a variation of quantitative nondivergence estimates for quasi-polynomial flows on the space of lattices.

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Audit summary

Audited against arXiv v1

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 Baker–Sprindzhuk-type strong-extremality theorem for complex analytic manifolds is correct under the stated real-linear independence hypothesis.

Theorem 1.1Correct

Complex analytic nondegenerate manifolds are strongly extremal

Pages 2–4 · Theorem 1.1 · arXiv:math/0210369v1

Real-linear independence of 1,f1,…,fn1,f_1,\ldots,f_n prevents any nonzero integer linear combination from vanishing identically. Complex analyticity supplies uniform good-function estimates, and quantitative nondivergence makes the very-well-multiplicatively-approximable limsup sets summable.

Full paper, version 1 ↗
Theorem 1.2 and the local formulationCorrect

The measure-theoretic extension

Sections 1 and 4 · arXiv:math/0210369v1

The local good/nonplanar hypotheses are precisely the properties used in the nondivergence argument. Covering the parameter domain by compact subballs and applying Borel–Cantelli proves the announced almost-everywhere conclusion without an exceptional boundary regime.

02Proofs2 reported findingsCorrect

The complex-analytic good-function estimates, exterior-algebra lower bounds, and Borel–Cantelli reduction are correct and complete.

Sections 2–3Correct and complete

Analytic functions satisfy the required uniform small-value bounds

Sections 2–3 · arXiv:math/0210369v1

Cauchy estimates and finite-order nonvanishing give a uniform power-law bound for sublevel sets of every relevant real and imaginary linear combination. The linear-independence hypothesis excludes identically zero combinations and makes the lower bound uniform on a smaller ball.

Section 4Correct and complete

Quantitative nondivergence implies strong extremality

Section 4 · proof of Theorem 1.1 · arXiv:math/0210369v1

The diagonal-flow lattice represents the multiplicative approximation inequalities, and every primitive exterior vector is controlled by the analytic estimates. Summing the resulting exponential measure bounds over discrete times and applying Borel–Cantelli gives the exact strong-extremality assertion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/0210369v1
Authors listed
Dmitry Kleinbock
Audit date
August 19, 2026
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