022. The weak inhomogeneous Duffin–Schaeffer conjecture

The weak inhomogeneous Duffin–Schaeffer conjecture

1 paper

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.

Papers 1

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

OpenAI · 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.

Read PDF Download Source

Cite

@misc{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026,
  author = {{OpenAI}},
  title = {{The Weak Inhomogeneous Duffin--Schaeffer Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf}{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026}},
  year = {2026}
}

The weak inhomogeneous Duffin–Schaeffer conjecture

Watch

Papers 1

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

Formal verification