MathAudit

Paper metadata

Original submission
October 16, 2026
Claimed MSC
11J83, 42A38, 28A78
arXiv categories
math.NT

Abstract

We determine the Fourier dimension of exactly approximation set for all dimensions. More precisely, let m,n≥1m,n\ge1, θ∈Rm\boldsymbolθ\in\mathbb R^m, and ψ:N→R+ψ:\mathbb N\to \mathbb R^+ be a non-increasing approximate function. Denote by Exactm,n(ψ,θ)\textrm{Exact}_{m,n}(ψ,\boldsymbolθ) the set of m×nm\times n matrices which are inhomogeneously ψψ-approximable with shift θ\boldsymbolθ, but not cψcψ-approximable for any fixed 0<c<10<c<1. We prove that dim⁡FExactm,n(ψ,θ)=2n1+τψ, whenever τψ>nm,\dim_{\text{F}} \textrm{Exact}_{m,n}(ψ,\boldsymbolθ)=\frac{2n}{1+τ_ψ},\ \text{whenever }τ_ψ>\frac{n}{m}, where τψτ_ψ is the lower order of ψψ at infinity.

AI-generated audit

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 23, 2026
01Statements1 reported findingSuspected unsupported statements

The claimed Fourier-dimension formula for every matrix dimension and arbitrary shift depends on a separation estimate that the paper proves only in the one-dimensional irrational-shift case. The other cases are attributed to an unavailable manuscript, so the full-dimensional theorem cannot be verified from the supplied argument and cited record.

Theorem 1.1Not independently verified

The general matrix and arbitrary-shift cases depend on an unverified estimate

PDF pages 4–8 and 12–18 · Proposition 2.1, Proposition 3.1, and proof of Theorem 1.1 · arXiv:2610.23680v1

Proposition 2.1 is the input that makes the dangerous resonant neighborhoods summable and hence puts full measure on the exact set. Although it is stated for all m,nm,n and all shifts, the paper explicitly proves only m=n=1m=n=1 with irrational shift. The remaining cases are said to be based on several results in reference [26], but that item is listed only as an unpublished manuscript and no precise substitute proof is supplied. Since Proposition 3.1 and the measure construction use the missing general estimate, the advertised all-dimensional conclusion remains unsupported here.

02Proofs2 reported findingsSuspected incomplete or incorrect proofs

Besides the unverified general form of Proposition 2.1, the final step proves Fourier decay only at integer torus frequencies although the paper's definition requires all real Euclidean frequencies. That second gap has a standard verified repair, but the central separation estimate still needs a citable full proof.

Unsupported central inputNot independently verified

Only one special case of Proposition 2.1 is proved

PDF pages 6–8 · Subsection 2.3 and Proposition 2.1 · arXiv:2610.23680v1

The text announces a statement covering both n=1n=1 and n≥2n\ge2, then says it will prove only m=n=1m=n=1 and irrational θ\theta. The omitted estimates are not routine dimension changes: they control overlaps between matrix resonant neighborhoods and all more accurate approximations, exactly the issue on which exactness of the limiting measure rests. A complete accessible source or proof is required before the central theorem can be certified.

Incomplete final inferenceSuspected incomplete as written · an apparently sound repair

Integer Fourier coefficients do not by themselves match the stated definition

PDF pages 2 and 17–18 · Definition 1 and Proposition 4.1 · arXiv:2610.23680v1

Proposition 4.1 establishes ∣μ^(k)∣≪(1+∣k∣)−α|\widehat\mu(k)|\ll(1+|k|)^{-\alpha} only for k∈Zmnk\in\mathbb Z^{mn} and immediately concludes a lower bound for the Euclidean Fourier dimension, whose definition on page 2 quantifies over every ξ∈Rmn\xi\in\mathbb R^{mn}. The conclusion can be repaired without changing the exponent: take a smooth finite partition of unity subordinate to coordinate boxes of the torus, choose a nonzero localized measure χμ\chi\mu, lift its support into Rmn\mathbb R^{mn}, and write ξ=k+δ\xi=k+\delta with δ\delta in a fixed bounded cube. Fourier-series convolution with χ(x)e−2πiδ⋅x\chi(x)e^{-2\pi i\delta\cdot x}, whose coefficients decay rapidly uniformly in δ\delta, transfers the integer decay to all real ξ\xi. This argument should be included.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.
Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper
arXiv:2610.23680v1
Authors listed
Yubin He, Bixuan Li, Baowei Wang
Audit date
October 23, 2026
  1. 01Establish identityMatch an authenticated scholarly identity to this paper.
  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
Recommended production method

Authenticate with ORCID, then require an exact arXiv match

MathAudit should accept the identity only when ORCID OAuth authenticates the claimant's iD and this exact arXiv paper appears in arXiv's public authority feed for that iD. A matching name alone is not sufficient.

ORCID OAuth and arXiv authority-record lookup are not connected in this local prototype.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

This demonstration does not send, store, or compare the address.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.