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.

Manuscript

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

September 25, 2026 · source · cite as \cite{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026}

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.

Result 022 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.