Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 022-01. The Weak Inhomogeneous Duffin–Schaeffer Conjecture
Manuscript map · lines 2915–2924
OpenAI@2026-10-07#rem:oai-p-022-01
Authors: OpenAI. Family 022.
Title: The Weak Inhomogeneous Duffin–Schaeffer Conjecture.
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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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