{"paper":{"title":"A note on the Duffin-Schaeffer conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liangpan Li","submitted_at":"2013-04-01T21:47:02Z","abstract_excerpt":"Given a sequence of real numbers $\\{\\psi(n)\\}_{n\\in\\mathbb{N}}$ with $0\\leq \\psi(n)<1$, let $W(\\psi)$ denote the set of $x\\in[0,1]$ for which $|xn-m|<\\psi(n)$ for infinitely many coprime pairs $(n,m)\\in\\mathbb{N}\\times\\mathbb{Z}$. The purpose of this note is to show that if there exists an $\\epsilon>0$ such that $\\sum_{n\\in\\mathbb{N}}\\psi(n)^{1+\\epsilon}\\cdot\\frac{\\varphi(n)}{n}=\\infty,$ then the Lebesgue measure of $W(\\psi)$ equals 1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0488","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}