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arXiv:2404.07752v2

Singular systems of linear forms over global function fields

Gukyeong Bang, Taehyeong Kim, Seonhee Lim

math.DSmath.NT11K6037P1537A44
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Paper metadata

Original submission
April 11, 2024
Latest revision
February 20, 2026 · version 2
Claimed MSC
11K60, 37P15, 37A44
arXiv categories
math.DS, math.NT
Paper license
CC BY 4.0 ↗
Comments
35 pages. Accepted for publication in Journal of Number Theory

Abstract

In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields.

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Audit summary

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 upper bound mn−mn/(m+n)mn-mn/(m+n) for singular systems over the stated class-number-one global function fields is correct. A formal error in the printed dimension argument has a verified local repair.

Theorem 1.1Correct

The singular-set dimension bound is correct

Pages 2 and 32–34 · Theorem 1.1 and Proposition 6.8 · arXiv:2404.07752v2

Dani correspondence sends singular matrices to trajectories that 11-escape on average. The covering theorem gives codimension mn/(m+n)mn/(m+n) for the escape set. Although the printed proof of Proposition 6.8 uses a false upper-box-dimension inequality for a countable union, replacing that line by countable stability of Hausdorff dimension proves the same estimate; the repair is detailed under Proofs.

Full paper, version 2 ↗
Proposition 6.8Correct

The escape-on-average estimate survives the repair

Pages 32–33 · Proposition 6.8 · arXiv:2404.07752v2

For each eventual-time set, the covers from Theorem 6.7 bound its Hausdorff dimension by mn−δ′mn/(m+n)mn-\delta' mn/(m+n) after t→∞t\to\infty. Hausdorff dimension of the countable union is the supremum of those dimensions, and then δ′→δ\delta'\to\delta gives the displayed proposition.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The Margulis-function and covering estimates are correct, but Proposition 6.8 contains an invalid upper-box-dimension step. A verified replacement using Hausdorff dimension repairs the argument.

Proposition 6.8Incorrect as written

Upper box dimension is applied incorrectly to a countable union

Page 33 · proof of Proposition 6.8, dimension chain after the eventual-time inclusion · arXiv:2404.07752v2

The proof bounds the upper box dimension of ⋃N0≥1⋂N≥N0Zx(Q,N,t,δ′)\bigcup_{N_0\geq1}\bigcap_{N\geq N_0}Z_x(Q,N,t,\delta') by the supremum of the upper box dimensions of its countably many pieces. Upper box dimension is not countably stable, so that inequality is false in general. Verified repair: write EN0=⋂N≥N0Zx(Q,N,t,δ′)E_{N_0}=\bigcap_{N\geq N_0}Z_x(Q,N,t,\delta') and use dim⁡H(⋃N0EN0)=sup⁡N0dim⁡HEN0≤sup⁡N0dim⁡‾BEN0\dim_H(\bigcup_{N_0}E_{N_0})=\sup_{N_0}\dim_H E_{N_0}\leq\sup_{N_0}\overline{\dim}_B E_{N_0}. Each EN0E_{N_0} is covered at the scales supplied by Theorem 6.7, giving the same limit bound. No later constant or quantifier changes.

Sections 3–6Correct and complete

The contraction and covering inputs are correct

Pages 5–32 · Sections 3–6 · arXiv:2404.07752v2

The wedge estimates and Hodge duality cover every exterior degree, the submodularity step produces the announced Margulis function, and the recursive integral estimate yields the stated number of ultrametric covering balls. These inputs are sufficient for the repaired final dimension argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2404.07752v2
Authors listed
Gukyeong Bang, Taehyeong Kim, Seonhee Lim
Audit date
August 19, 2026
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