{"paper":{"title":"The Duffin-Schaeffer type conjectures in various local fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liangpan Li","submitted_at":"2013-12-30T22:02:41Z","abstract_excerpt":"This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\\lambda(\\cap_{m=1}^{\\infty}\\cup_{n=m}^{\\infty}{\\mathcal E}_n)=1$ if and only if $\\sum_n\\lambda({\\mathcal E}_n)=\\infty$, where $\\lambda$ denotes the Lebesgue measure on $\\mathbb{R}/\\mathbb{Z}$, \\[ {\\mathcal E}_n={\\mathcal E}_n(\\psi)=\\bigcup_{m=1 \\atop (m,n)=1}^n\\big(\\frac{m-\\psi(n)}{n},\\frac{m+\\psi(n)}{n}\\big), \\] $\\psi$ is any non-negative arithmetical function. Instead of studying $\\cap_{m=1}^{\\infty}\\cup_{n=m}^{\\infty}{\\mathcal E}_n$ we introduce an even fundamental object $\\cup_{n=1}^{\\infty}{\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0035","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"}