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arXiv:2405.11274v2

Hausdorff dimension of singular vectors in function fields

Noy Soffer Aranov, Taehyeong Kim

math.NTmath.DS11J1311K5537A17
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Paper metadata

Original submission
May 18, 2024
Latest revision
December 5, 2024 · version 2
Claimed MSC
11J13, 11K55, 37A17
arXiv categories
math.NT, math.DS
Paper license
CC BY 4.0 ↗
Comments
25 pages, Minor errata were corrected

Abstract

We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of ε\varepsilon-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier [Duke Math. J. 165 (2016), 2273--2329].

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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.

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Detailed mathematical audit

Generated August 19, 2026
01Statements3 reported findingsCorrect

The singular-set dimension formula and the effective upper and lower estimates for Dirichlet-improvable vectors are correct, subject to one local correction of the displayed range of the lower estimate.

Theorem 1.2(1)Correct

The singular-vector dimension is correct

Pages 3 and 18–20 · Theorem 1.2(1) and Corollary 6.14 · arXiv:2405.11274v2

The lower construction gives dimension at least d2/(d+1)d^2/(d+1) while the cited function-field upper bound gives the reverse inequality. The construction sends its improvement parameters to zero, so every limit point is singular and the two bounds apply to the same set.

Full paper, version 2 ↗
Theorem 1.2(2)Minor formal correction

The lower-bound range must match Corollary 6.13

Page 3 · Theorem 1.2(2); page 19 · Corollary 6.13 · arXiv:2405.11274v2

The main theorem introduces the two-sided estimate with ‘for any ε>0\varepsilon>0’, but the proved lower estimate assumes 0<ε<((q−1)/q)1/(d−1)0<\varepsilon<((q-1)/q)^{1/(d-1)}. Outside that range the factor (q−1)/q−εd−1(q-1)/q-\varepsilon^{d-1} changes sign, and for sufficiently large ε\varepsilon even the logarithm in the displayed lower bound need not be real. Replace the range attached to the lower inequality by 0<ε<((q−1)/q)1/(d−1)0<\varepsilon<((q-1)/q)^{1/(d-1)}; the upper inequality remains valid in its stated range. This is a local range correction and does not affect the singular-dimension result or any later application.

Theorem 1.2(2), upper estimateCorrect

The upper estimate is correct

Pages 3 and 11–13 · Theorem 1.2(2) and Corollary 5.4 · arXiv:2405.11274v2

The Farey-lattice characterization produces a self-similar cover of the Dirichlet-improvable set. The successor count and contraction ratio give the displayed logarithmic correction after solving the covering inequality, with constants uniform over the nodes.

02Proofs2 reported findingsCorrect

The best-approximation, Farey-lattice, covering, and lower-fractal arguments are correct and complete in the parameter ranges actually used.

Sections 4–5Correct and complete

The characterization and upper cover are correct

Pages 7–13 · Sections 4–5 · arXiv:2405.11274v2

The ultrametric best-approximation identities give the exact Dirichlet-improvability criterion, and the Farey-lattice point count supplies the covering exponent used in the upper-bound corollary.

Section 6Correct and complete

The lower constructions establish their stated corollaries

Pages 14–20 · Section 6 · arXiv:2405.11274v2

Within 0<ε<((q−1)/q)1/(d−1)0<\varepsilon<((q-1)/q)^{1/(d-1)}, the child mass is positive, separation is strict, and the lower-dimension theorem yields Corollary 6.13. The varying-parameter construction has improvement parameters tending to zero and its mass estimates approach d2/(d+1)d^2/(d+1), proving the singular lower bound.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2405.11274v2
Authors listed
Noy Soffer Aranov, Taehyeong Kim
Audit date
August 19, 2026
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