The inhomogeneous Duffin–Schaeffer conjecture
The inhomogeneous Duffin–Schaeffer conjecture concerns approximation of a real number x by rationals with a fixed additive shift γ: for an arbitrary approximation function ψ, divergence of Σ φ(q)ψ(q)/q should imply that |qx − a − γ| < ψ(q) has infinitely many solutions for almost every x. The homogeneous Duffin–Schaeffer conjecture (γ = 0, with reduced fractions) was proved by Koukoulopoulos and Maynard in 2020. In the inhomogeneous setting two versions are distinguished: the weak version, in which the integer a is arbitrary, and the reduced version, which requires a to be coprime to q. A manuscript dated 25 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims the weak version for every real shift. In the same days, two groups of human authors posted counterexamples to the reduced version. This page summarizes the published claims and lists related public work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the manuscript's introduction: the fixed-shift weighted question appears as Question 1.2 of Yu, as Conjecture 1.22 of Chow and Technau under the name weak inhomogeneous Duffin–Schaeffer conjecture, and as Conjecture 2 of Beresnevich, Hauke and Velani. The distinction between the weak and the coprime-numerator formulations is already explicit in work of Ramírez. Beresnevich, Hauke and Velani proved the weak conjecture for every rational shift. Yu's Fourier approach and Chow and Technau's results treat further cases under additional restrictions on the shift or the approximation function.
The original result
- Claim. Let γ be any real number and ψ : ℕ → [0, ∞) any finite-valued function. If Σq φ(q)ψ(q)/q diverges, then for Lebesgue-almost every real x the inequality ‖qx − γ‖ < ψ(q) holds for infinitely many positive integers q (Theorem 1.1). Here ‖·‖ is the distance to the nearest integer, so the numerator is unrestricted; no monotonicity of ψ and no Diophantine condition on γ is assumed. The manuscript also derives a Hausdorff-measure consequence through the mass transference principle.
- Source. The manuscript The weak inhomogeneous Duffin–Schaeffer conjecture, dated 25 September 2026 (repository commit
1581d4c). - Formalization. The repository has no Lean scope note for this family (checked 16 October 2026).
- Relation to the counterexamples, as stated in the manuscript. The introduction cites both papers below and states that the coprime-numerator assertion is false in general, whereas its theorem concerns the weak formulation.
Related work by others
| Date | Work | Relation | Public claim | Public status (checked 16 October 2026) |
|---|---|---|---|---|
| 2026-09-25 | Manuel Hauke-Treuer, James Maynard and Andrew Pollington, arXiv:2609.30921 (v1), Counterexamples to the inhomogeneous Duffin–Schaeffer Conjecture | Counterexamples to the reduced version (cited by the manuscript) | Approximation functions for which the inhomogeneous analogue of the Duffin–Schaeffer conjecture, with reduced fractions, fails, for all non-zero rational shifts and certain Liouville numbers. | arXiv preprint, version 1. |
| 2026-09-25 | Yubin He and Lingmin Liao, arXiv:2609.30870 (v1; v2 of 9 October), Counterexamples to the inhomogeneous Duffin–Schaeffer conjecture for a residual set of shifts | Counterexamples to the reduced version (cited by the manuscript) | For every non-zero rational shift, an approximation function with divergent weighted sum for which the coprime inhomogeneous inequality has only a null set of solutions; a modification gives failure for a residual set of shifts. | arXiv preprint, version 2. The acknowledgements state that the counterexample was discovered with the assistance of OpenAI's GPT-5.6 Sol model, with all arguments and references verified by the authors, and that the authors later learned that Andrew Pollington had announced a counterexample at a conference in York more than a year earlier. |
Notes
- Dates are those shown by arXiv and by the manuscript. The list covers sources found up to the date above; it makes no priority or completeness claim.