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The Weak Inhomogeneous Duffin–Schaeffer Conjecture
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Abstract
We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued $\psi:\mathbb N\to[0,\infty)$, divergence of $\sum_{q\ge1}\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|\lt \psi(q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and $\|\cdot\|$ is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- Manuel Hauke and Felipe A. Ramírez, The Duffin-Schaeffer conjecture with a moving target (arXiv:2407.05344, July 2024). Gives a one-dimensional counterexample when the inhomogeneous parameter varies with the denominator, so a one-dimensional proof must use that the shift is fixed; Theorem 1.1 is stated for a fixed shift $\gamma$.
- Stefan M. Hesseling and Felipe A. Ramirez (appendix by Manuel Hauke), Duffin–Schaeffer examples, real residue systems, and Bohr-set primes (arXiv:2602.23511, February 2026, revised 2026-03-04). Constructs $\psi$ whose inhomogeneous limsup set is null for shifts in one prescribed countable set and full for shifts in another (under a rational-independence condition); relevant to the paper's discussion of shift-dependent counterexamples and to its null set depending on $\gamma$.
- Demi Allen, Manuel Hauke-Treuer and Felipe A. Ramírez, The inhomogeneous Khintchine Theorem in dimension two (arXiv:2605.19582, May 2026). Proves the inhomogeneous Khintchine theorem in dimension two without monotonicity and notes that monotonicity is needed in dimension one; background for the paper's use of the totient-weighted sum to drop monotonicity in dimension one.
- Bo Tan and Qing-Long Zhou, Fourier Dimension in Inhomogeneous Duffin–Schaeffer Conjecture (arXiv:2604.13868, revised 2026-04-16). Studies Fourier dimension of shifted approximation limsup sets with numerator coprimality conditions; a possible comparison for the Hausdorff-measure conclusion of Corollary 1.2.
Digests and expositions
- Latent Seminar, Weak inhomogeneous Duffin–Schaeffer approximation (2026-10-06). Explains the fixed-shift quantifiers, unrestricted numerators, finite-block construction, prime-step matching and Hausdorff-measure consequence, stating that it reviews statements and selected proof mechanisms rather than verifying the complete proof. [family-level]
- Perplexity / Orply, Weighted Divergence Guarantees Infinitely Many Inhomogeneous Approximations (2026-10-08). Summarizes the manuscript's weighted divergence criterion and distinguishes its local prime-step mass identity from the global overlap estimates needed by the proposed proof. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 022
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026,
author = {{OpenAI}},
title = {{The Weak Inhomogeneous Duffin--Schaeffer Conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf}{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026}},
year = {2026}
}Family 022: The weak inhomogeneous Duffin–Schaeffer conjecture
Proves that for every real shift γ and finite-valued $\psi:\mathbb N\to[0,\infty)$, divergence of $\sum_q\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|\lt \psi(q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.
This is the only paper in this family.