{"paper":{"title":"An almost sharp quantitative version of the Duffin-Schaeffer conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daodao Yang, Dimitris Koukoulopoulos, James Maynard","submitted_at":"2024-04-22T23:54:58Z","abstract_excerpt":"We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $\\psi:\\mathbb{N}\\to[0,1/2]$ be a function such that the series $\\sum_{q=1}^\\infty \\varphi(q)\\psi(q)/q$ diverges. In addition, given $\\alpha\\in\\mathbb{R}$ and $Q\\geqslant1$, let $N(\\alpha;Q)$ be the number of coprime pairs $(a,q)\\in\\mathbb{Z}\\times\\mathbb{N}$ with $q\\leqslant Q$ and $|\\alpha-a/q|<\\psi(q)/q$. Lastly, let $\\Psi(Q)=\\sum_{q\\leqslant Q}2\\varphi(q)\\psi(q)/q$, which is the expected value of $N(\\alpha;Q)$ when $\\alpha$ is uniformly chosen from $[0, 1]$. We prove that $N(\\a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.14628","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.14628/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}