{"id":"openai-math/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026","tags":["number-theory","diophantine-approximation"],"score":0,"title":"The Weak Inhomogeneous Duffin–Schaeffer Conjecture","domain":"Number theory","family":"022","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"The weak inhomogeneous Duffin–Schaeffer conjecture"}