The weak inhomogeneous Duffin–Schaeffer conjecture

What OpenAI claims

Proves that for every real shift γ and finite-valued ψ:ℕ → [0,∞), divergence of ∑_qϕ(q)ψ(q)/q implies ‖qx-γ‖ < ψ(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.

Paper

1. 0,∞), divergence of ∑_qϕ(q)ψ(q)/q implies ‖qx-γ‖ < ψ(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.  [The Weak Inhomogeneous Duffin–Schaeffer Conjecture

   25 September 2026 · Preprint, not peer reviewed · PDF

   We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued ψ:ℕ → [0,∞), divergence of ∑_{q ≥ 1}ϕ(q)ψ(q)/q implies ‖qx-γ‖ < ψ(q) for infinitely many q, for almost every x. Here ϕ is Euler's totient and ‖·‖ is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.