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The weak inhomogeneous Duffin-Schaeffer conjecture

Koukoulopoulos and Maynard (2020) proved the Duffin-Schaeffer conjecture: for ψ:N→[0,∞)\psi:\mathbb{N}\to[0,\infty) with ∑qϕ(q)ψ(q)/q=∞\sum_q\phi(q)\psi(q)/q=\infty, almost every xx has infinitely many reduced fractions a/qa/q with ∣x−a/q∣<ψ(q)/q|x-a/q|<\psi(q)/q. For a fixed shift γ\gamma, unweighted divergence does not suffice without monotonicity (Ramirez; Chow-Hauke-Pollington-Ramirez), and the coprime-numerator inhomogeneous version is false. The weak inhomogeneous conjecture (Yu 2021, Question 1.2; Chow-Technau 2024, Conjecture 1.22; Beresnevich-Hauke-Velani 2024, Conjecture 2), proved for rational shifts by Beresnevich-Hauke-Velani, asks: for every fixed γ∈R\gamma\in\mathbb{R} and every ψ\psi with ∑qϕ(q)ψ(q)/q=∞\sum_q\phi(q)\psi(q)/q=\infty, does ∥qx−γ∥<ψ(q)\|qx-\gamma\|<\psi(q) hold for infinitely many qq, for almost every xx?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Metric Diophantine approximation
Posed by
H. Yu, J. Number Theory (2021), Question 1.2; named by Chow and Technau (2024, Conjecture 1.22); also Beresnevich, Hauke and Velani (2024), Conjecture 2
Year posed
2021
Years open
5y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every fixed γ∈R\gamma\in\mathbb{R} and finite-valued ψ:N→[0,∞)\psi:\mathbb{N}\to[0,\infty) with ∑qϕ(q)ψ(q)/q=∞\sum_q\phi(q)\psi(q)/q=\infty, almost every xx satisfies ∥qx−γ∥<ψ(q)\|qx-\gamma\|<\psi(q) for infinitely many qq; the null set may depend on γ\gamma and ψ\psi. Corollary 1.2 gives the Hausdorff-measure analogue via mass transference. The arithmetic core adapts the Koukoulopoulos-Maynard common-pivot method (as simplified by Green-Walker). It proves no convergence converse, no quantitative asymptotic, no coprime-numerator statement, and no result uniform in the shift.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated in the three sources the manuscript cites; the proof was not refereed. The statement covers every real shift, including irrational ones, with arbitrary nonmonotone ψ\psi and unrestricted numerators. The paper notes that the coprime-numerator version is false (Hauke-Treuer-Maynard-Pollington; He-Liao, both 2026), so this entry concerns only the weak form. No Lean formalization is supplied for this family.

Sources

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