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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.0035","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-30T22:02:41Z","cross_cats_sorted":[],"title_canon_sha256":"4d62942e281bb11df7db8b1c663ef7cf5d513abf475b3a162a16afae87475980","abstract_canon_sha256":"4c23a41a9d19c3f3474c4ece02d5e2504a422b6460abde557a08c3d1d62f36c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:15.518243Z","signature_b64":"eJcA6uuObI41Qz7OWi381X+eEdqYRyDFSucYH1S6dwgsgCkXCqt2HQhyBqUr9oDH3RrUYVCKAqZLgY3u5aYyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d13d23b57a716cd3f640b25e9ff9abc40b11e41bf4c66824865a43638572485d","last_reissued_at":"2026-05-18T01:15:15.517568Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:15.517568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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