arXiv:2404.15123v2

Proving the Duffin-Schaeffer conjecture without GCD graphs

Manuel Hauke, Santiago Vazquez Saez, Aled Walker

math.NT11J8311K6060F2011J71

Paper metadata

Original submission: April 23, 2024
Latest revision: July 10, 2026 · version 2
Claimed MSC: 11J83, 11K60, 60F20, 11J71
arXiv categories: math.NT
Paper license: CC BY-NC-ND 4.0 ↗
Comments: Minor edits due to referee reports; to appear in Trans. Amer. Math. Soc

Abstract

We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the $(\log Ψ(N))^{-C}$ error-term of Aistleitner-Borda and the first named author to $\exp(-(\log Ψ(N))^{\frac{1}{2} - \varepsilon})$ .

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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 15, 2026

01Statements2 reported findingsCorrect

The bilinear estimate of Theorem 1.7, its deduction of the Duffin–Schaeffer theorem, and the quantitative almost-sure error term are supported. Two local notation and endpoint defects in Section 6 have unique corrections and do not affect any application or overall status.

Theorem 1.7Correct

The bilinear exceptional-pair estimate is correct

Pages 5–17 · Theorem 1.7 and Propositions 2.1–2.2 · arXiv:2404.15123v2

Minimality in the prime support gives the valuation-slice estimate of Lemma 3.1. Lemma 3.2 then concentrates every prime's valuation measure in one diagonal cell, with a summable loss because both exponents are strictly greater than $1$ . Removing the exceptional mass produces a common valuation center $N$ . On that structured remainder, the four possible off-center valuation patterns and the two anatomy lemmas give the square-root estimate $\mu^{f,g}_{\psi,\theta}(E')\ll\bigl(\mu^f_\psi(V')\mu^g_\theta(W')e^{-25Ct}\bigr)^{1/2}.$ The final small-versus-large mass split upgrades the exponent to $1/2+\varepsilon$ with the stated uniform constant.

Corollary 1.3Correct

The quantitative Duffin–Schaeffer estimate is correct

Pages 3 and 17–23 · Corollary 1.3 and Section 6 · arXiv:2404.15123v2

The overlap sum is partitioned by the size of the gcd parameter and by the large-prime anatomy statistic. Theorem 1.7 controls the small-gcd and anatomy-heavy pieces, while the cited overlap estimate gives the asymptotic main term on the remaining pairs. This yields the variance bound with error $\exp\bigl(-(\log\Psi(N))^{1/2-\varepsilon}\bigr)$ . Chebyshev's inequality on the level sequence for $\Psi$ , followed by Borel–Cantelli and monotone interpolation, gives the asserted almost-sure estimate.

02Proofs3 reported findingsCorrect

The minimal-counterexample reduction, valuation concentration, anatomy estimates, and second-moment argument are correct and complete. One duplicated variable is a typo, and Proposition 6.3 should exclude the undefined endpoint $A=0$ ; both are yellow findings with no downstream mathematical effect.

Page 18 definition of $D(n,m)$Typo

The second argument is accidentally duplicated

Page 18 · paragraph preceding Proposition 6.2 · arXiv:2404.15123v2

The paper prints $D(n,m)=\frac{\max\{m\psi(n),m\psi(n)\}}{\gcd(n,m)}.$ The second entry must be $n\psi(m)$ , as in Definition 1.6 and every earlier occurrence: $D(n,m)=\frac{\max\{m\psi(n),n\psi(m)\}}{\gcd(n,m)}.$ The words “as before,” the symmetry in $(n,m)$ , and all subsequent uses determine this correction uniquely.

Full paper, version 2 ↗

Proposition 6.3Minor formal correction

The parameter $A=0$ must be excluded

Page 19 · statement and proof of Proposition 6.3 · arXiv:2404.15123v2

The proposition begins with $s,t,A\geq0$ , but its defining condition contains $L_t(n,m)\geq1/A$ and its proof sets $C=1/A$ . Both expressions are undefined at $A=0$ . Replace the displayed range for this parameter by $A>0$ . Every use in the proof of Corollary 6.1 has a strictly positive $A$ , so the correction is local and changes no conclusion.

Full paper, version 2 ↗

Sections 3–6Correct and complete

Central proof chain

Pages 8–23 · Lemmas 3.1–4.2, Proposition 2.2, and Corollary 6.1 · arXiv:2404.15123v2

The conjugate-exponent calculations in the valuation concentration lemma, the summability over large primes, and the four structured valuation cases are consistent. The multiplicative-function hypotheses supply exactly the divisor bounds used in Lemma 4.2. In Section 6, each exceptional-pair class is covered, the overlap estimate is invoked within its parameter range, and the resulting decay is sufficient for the level-sequence Borel–Cantelli argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2404.15123v2
Authors listed: Manuel Hauke, Santiago Vazquez Saez, Aled Walker
Audit date: August 15, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. 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P(delta)

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Proving the Duffin-Schaeffer conjecture without GCD graphs

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The bilinear exceptional-pair estimate is correct

The quantitative Duffin–Schaeffer estimate is correct

The second argument is accidentally duplicated

The parameter A=0A=0A=0 must be excluded

Central proof chain

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The parameter $A=0$ must be excluded