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The weak inhomogeneous Duffin-Schaeffer conjecture

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.

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The weak inhomogeneous Duffin-Schaeffer conjecture · Weak inhomogeneous Duffin-Schaeffer

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Beresnevich, Hauke, Velani, Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer (arXiv:2406.19198) evidence for this event

VibeMathed record: The weak inhomogeneous Duffin-Schaeffer conjecture evidence for this event

The weak inhomogeneous Duffin-Schaeffer conjecture evidence for this event

The weak inhomogeneous Duffin-Schaeffer conjecture parent of this event

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. parent of this event

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