OpenAI · 022-01 · 2026-09-25 · Number theory

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026

Family 022. The weak inhomogeneous Duffin–Schaeffer conjecture.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves that for every real shift $γ$ and finite-valued $ψ : N \to [0, \infty)$ , divergence of $\sum_{q} ϕ (q) ψ (q) / q$ implies $∥ q x - γ ∥ < ψ (q)$ 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.

We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued $ψ : N \to [0, \infty)$ , divergence of $\sum_{q \geq 1} ϕ (q) ψ (q) / q$ implies $∥ q x - γ ∥ < ψ (q)$ 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.

Preprint source and Lean scope

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