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Paper metadata

Original submission
March 8, 2025
Latest revision
June 3, 2026 · version 2
Claimed MSC
11J61, 11J13, 11J83, 37A17, 22E40
arXiv categories
math.NT, math.DS
Comments
Final accepted version

Abstract

In this article, we determine the Hausdorff dimension of the set of exactly ψψ-approximable vectors over local fields of positive characteristic. This result is the function field analogue of a recent theorem of Bandi and de Saxcé in the real setting, and extends the main theorem of Zhang to higher dimensions. Our approach adapts the method of Bandi and de Saxcé to the ultrametric setting, enabling us to overcome difficulties arising from the failure of the well-separatedness property for rational functions in higher dimensions.

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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 18, 2026
01Statements2 reported findingsContains unsupported statements

The Hausdorff-dimension theorem is supported, including the zero-dimensional infinite-order case. The separate nonemptiness theorem is not verified for approximation functions of infinite lower order because its proof explicitly treats only λψ<∞\lambda_\psi<\infty.

Theorem 1.3Correct

Hausdorff dimension of the exact-approximation set

Pages 2–14 · Theorem 1.3 and Sections 2–4 · arXiv:2503.06110v2

The limsup cover gives the upper bound (n+1)/λψ(n+1)/\lambda_\psi. The lattice-template construction supplies exact rather than merely infinitely-often approximation, and the Cantor measure gives the matching lower bound when λψ<∞\lambda_\psi<\infty. When λψ=∞\lambda_\psi=\infty, the formula reads zero and the upper bound already gives dimension zero, so no positive lower estimate is required for this theorem.

Theorem 1.5Not able to verify

Nonemptiness is not proved for infinite lower order

Pages 2 and 14 · statement and proof of Theorem 1.5 · arXiv:2503.06110v2

The hypotheses allow λψ=∞\lambda_\psi=\infty, for example ψ(qk)=q−k2\psi(q^k)=q^{-k^2} after monotone interpolation on the allowed values. The entire printed proof is: when λψ<∞\lambda_\psi<\infty, Theorem 1.3 gives positive dimension and hence nonemptiness. It says nothing about λψ=∞\lambda_\psi=\infty. Earlier template and Cantor constructions may contain ingredients for a separate existence proof, but the manuscript does not show that they retain a branch and satisfy exactness in this regime, and no independent complete proof was obtained here.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The proof of Theorem 1.5 omits an allowed case, λψ=∞\lambda_\psi=\infty. The dimension proof is otherwise complete, and the omission does not change Theorem 1.3.

Proof of Theorem 1.5Incomplete as written

The infinite-order case is absent

Page 14 · final proof · arXiv:2503.06110v2

Positive Hausdorff dimension proves nonemptiness only when (n+1)/λψ>0(n+1)/\lambda_\psi>0. For λψ=∞\lambda_\psi=\infty, Theorem 1.3 yields dimension zero, which is compatible with either an empty or a nonempty set. A repair must give a direct nested-construction argument in the super-polynomial regime or add the hypothesis λψ<∞\lambda_\psi<\infty to Theorem 1.5.

Definition of lower orderTypo

The lower order is printed as if it were a function of xx

Page 2 · definition preceding Theorem 1.3 · arXiv:2503.06110v2

The manuscript writes λψ(x):=lim inf⁡x→∞(−log⁡ψ(x))/log⁡x\lambda_\psi(x):=\liminf_{x\to\infty}(-\log\psi(x))/\log x. The limit is a scalar and every later formula uses λψ\lambda_\psi without an argument. Removing the spurious (x)(x) is the unique correction and has no mathematical effect.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2503.06110v2
Authors listed
Aratrika Pandey
Audit date
August 18, 2026
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