MathAudit

Paper metadata

Original submission
October 20, 2026
Claimed MSC
11J83, 11J86, 11K60, 11K70, 28A80
arXiv categories
math.NT, math.DS
Paper license
CC BY 4.0 ↗

Abstract

We investigate the approximation of elements of the middle-third Cantor set by dyadic rationals, and natural extensions. We prove a conjecture of Velani, completing the metric theory for the Cantor measure. Our key method enables a quantitative study of the incommensurability of binary and ternary expansions of integers, initiated by Furstenberg. The method has other applications, e.g. ∑k∣μ^(2k)∣σ<∞\sum_{k} |\hatμ(2^k)|^σ< \infty for any σ>0σ> 0, where μμ is the Cantor measure.

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
01Statements2 reported findingsNo statement issues found

The missing-digit Khintchine theorem, averaged Fourier-decay estimate, and quantitative normality theorem are supported. A handful of copied factors and indices in the printed Fourier calculations have unique local corrections.

Theorems 1.4 and 3.1No issue found

Averaged Fourier decay supplies both halves of the metric theorem

PDF pp. 3–5 and 10–25 · arXiv:2610.28839v1

Stewart's digit-change estimate controls the qualified frequencies uniformly over a dyadic block, while rapid decay handles the remaining frequencies. The convergent half follows from the first-moment bound and Borel–Cantelli. The divergent half combines the close-scale Fourier estimate with localization and the Lebesgue quasi-independence bound.

Theorem 1.8No issue found

The discrepancy estimate gives the stated quantitative normality

PDF pp. 5–6 and 26–30 · arXiv:2610.28839v1

The exponential-sum estimate is uniform for the required digit blocks. Erdős–Turán and the dyadic decomposition then give the displayed discrepancy rate outside the exceptional set, with the parameter choices satisfying the stated inequalities.

02Proofs4 reported findingsNo proof issues found

The central Fourier, Borel–Cantelli, localization, and discrepancy arguments are complete after the uniquely determined notation repairs recorded below.

Proof of Theorem 3.1Suspected typo

Restore the missing frequency variable in two Fourier terms

PDF p. 17 · final page of the proof of Theorem 3.1 · arXiv:2610.28839v1

Stewart's bound is applied to f(pK+h+Δt/qℓ)f(p^{K+h+\Delta}t/q^\ell), but the displayed line omits tt. The final qualified sum likewise must contain f(pK+ht/qℓ)f(p^{K+h}t/q^\ell) rather than f(pK+h/qℓ)f(p^{K+h}/q^\ell). The summation variable and the preceding formula uniquely determine both corrections.

Equation (5.5)Suspected typo

Use the same Q3Q^3 cutoff in both halves of the frequency split

PDF pp. 21–22 · proof of (5.5) · arXiv:2610.28839v1

The split is announced at max⁡{∣t∣,∣t′∣}≥Q100\max\{|t|,|t'|\}\geq Q^{100}, but the two cases actually proved are max⁡{∣t∣,∣t′∣}≤Q3\max\{|t|,|t'|\}\leq Q^3 and max⁡{∣t∣,∣t′∣}≥Q3\max\{|t|,|t'|\}\geq Q^3. Replace Q100Q^{100} by Q3Q^3 in the announcing sentence. This removes the apparent uncovered range and leaves the estimates unchanged.

Section 5.2Suspected typo

Normalize the nonzero Fourier coefficients inside the factored average

PDF p. 23 · Fourier expansion of ν(f)\nu(f) · arXiv:2610.28839v1

With the paper's Fourier convention the exact identity is ν(f)=λ(f)+∑t≠0f^(pk′−k−ht)ν^(−pk′−k−ht).\nu(f)=\lambda(f)+\sum_{t\neq0}\widehat f(p^{k'-k-h}t)\widehat\nu(-p^{k'-k-h}t). If λ(f)\lambda(f) is factored out as printed, each coefficient in the parenthesis must be divided by λ(f)\lambda(f). The following bound uses precisely this normalized version, so the correction is mechanical.

Section 5.4Suspected typo

The higher-dimensional citation has a copied dimension

PDF p. 25 · Lebesgue quasi-independence input · arXiv:2610.28839v1

The sentence introduces Tan–Zhou's higher-dimensional result but then says it is used only in the case d=2d=2. The theorem being proved and the cited higher-dimensional lemma require and supply the case d≥2d\geq2; replace d=2d=2 by d≥2d\geq2.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.28839v1
Authors listed
Sam Chow, Han Yu
Audit date
October 23, 2026
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