The weak inhomogeneous Duffin–Schaeffer conjecture
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves that for every real shift γ and finite-valued \psi:\mathbb N→[0,∞), divergence of \sum_q\phi(q)\psi(q)/q implies \|qx-\gamma\|< \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.
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MANUSCRIPT 01
The Weak Inhomogeneous Duffin–Schaeffer Conjecture
We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued \psi:\mathbb N→[0,∞), divergence of \sum_{q\ge1}\phi(q)\psi(q)/q implies \|qx-\gamma\|< \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…
Full source abstract
We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued \psi:\mathbb N→[0,∞), divergence of \sum_{q\ge1}\phi(q)\psi(q)/q implies \|qx-\gamma\|< \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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