MathAudit

Paper metadata

Original submission
October 2, 2026
Claimed MSC
11J83, 28A78
arXiv categories
math.NT

Abstract

We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let ψ:N→(0,∞)ψ:\N\to(0,\infty) be a non-increasing approximation function and let τψτ_ψ denote its lower order. For m,n≥1m,n\ge1 and θ∈Rm\boldsymbolθ\in\mathbb R^m, denote Exact_m,n(ψ,boldsymbolθ) the set of m×nm\times n matrices that are ψψ-approximable with shift θ\boldsymbolθ, but are not cψcψ-approximable for any 0<c<10<c<1. It is shown that hdim Exact_m,n(ψ,boldsymbolθ)=m(n-1)+m+n/1+τ_ψ,\ \ \ textfor\ τ_ψ>n/m. We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces {p+boldsymbolθ/q: (p, q)in Z^m+1},\ \ \ bigg{big{Ain Mat_m, n(R): Aq-p-boldsymbolθ=0big}: (p, q)in Z^m+nbigg}.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 24, 2026
01Statements1 reported findingNo statement issues found

For every pair of positive dimensions and every inhomogeneous shift, the stated Hausdorff-dimension formula for exactly approximable matrices follows when the lower order of the approximation function exceeds the Dirichlet threshold.

Theorem 1.1No issue found

The upper and lower dimension bounds match

PDF pp. 3–4, 20–22, 33–34 · arXiv:2610.03442v1

The standard dimension formula for the ambient well-approximable limsup set supplies the upper bound. For one denominator variable, the authors construct nested exact cubes that avoid every smaller target eventually and retain the required number and separation of children. For several denominator variables, the same construction is carried out on a positive-measure set of vertical fibres and the slicing lemma contributes the remaining m(n−1)m(n-1) dimensions.

02Proofs2 reported findingsNo proof issues found

The regular-scale reduction, averaged collision counts, goodness estimates, separation bounds, Cantor mass calculation, fibre exceptional-set construction, and final slicing argument were checked through their principal exponents and quantifiers.

Sections 3–7No issue found in the reviewed proof

The one-column Cantor construction has the claimed dimension

PDF pp. 7–22 · arXiv:2610.03442v1

Prime denominators make the congruence parametrization injective. Good parent cubes leave a positive proportion of exact children after small-denominator deletion; the rational and irrational shifts are separately controlled for intermediate denominators. Close resonant points and children with excessive later overlap form negligible families. The recursive growth choices then put the child counts, gaps, and side lengths in the two-scale mass-distribution lemma and give the lower bound (m+1)/(1+τψ)(m+1)/(1+\tau_\psi).

Sections 8–12No issue found in the reviewed proof

Averaging over fibres preserves the counting exponents

PDF pp. 22–34 · arXiv:2610.03442v1

Haar invariance gives the exact mean count for each affine congruence, and linear independence makes the exact and dangerous events independent. Summation over denominator vectors is finite precisely under mτψ>nm\tau_\psi>n. Markov estimates remove summable exceptional fibre sets, leaving one positive-measure set on which every construction stage works simultaneously. Fibres with coincident resonant parametrizations form a countable union of Haar-null affine subtori and can be absorbed into the same exceptional set. The fibre Cantor bound and slicing lemma give m(n−1)+(m+n)/(1+τψ)m(n-1)+(m+n)/(1+\tau_\psi).

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.03442v1
Authors listed
Yubin He, Bixuan Li, Baowei Wang
Audit date
October 24, 2026
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