{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:JAKQKRECMWMUDVXGSKZ3OHGCIJ","short_pith_number":"pith:JAKQKREC","schema_version":"1.0","canonical_sha256":"4815054482659941d6e692b3b71cc242451addcb6a7e9eb78e39371cf3d401f0","source":{"kind":"arxiv","id":"2404.14628","version":2},"attestation_state":"computed","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"},"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":"2404.14628","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-22T23:54:58Z","cross_cats_sorted":[],"title_canon_sha256":"5469190d2de83bded704a4e76556c790873ebfa85f56c9b4976c1fff4a8c38a8","abstract_canon_sha256":"7c84224ce6254119ccc2dae02681685f2b8dc69ac8c0d05ce7e2f9880369c368"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:09:21.893163Z","signature_b64":"qgeYt5fneAqDlH+Tu0pKXoqpWR83MJq4Z0YUaaX30IiEuIF7hGMVATL001MBWitGhMRoivokGjIitJwu5zZfBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4815054482659941d6e692b3b71cc242451addcb6a7e9eb78e39371cf3d401f0","last_reissued_at":"2026-07-05T09:09:21.892564Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:09:21.892564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2404.14628","created_at":"2026-07-05T09:09:21.892647+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.14628v2","created_at":"2026-07-05T09:09:21.892647+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.14628","created_at":"2026-07-05T09:09:21.892647+00:00"},{"alias_kind":"pith_short_12","alias_value":"JAKQKRECMWMU","created_at":"2026-07-05T09:09:21.892647+00:00"},{"alias_kind":"pith_short_16","alias_value":"JAKQKRECMWMUDVXG","created_at":"2026-07-05T09:09:21.892647+00:00"},{"alias_kind":"pith_short_8","alias_value":"JAKQKREC","created_at":"2026-07-05T09:09:21.892647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.08901","citing_title":"A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard","ref_index":50,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ","json":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ.json","graph_json":"https://pith.science/api/pith-number/JAKQKRECMWMUDVXGSKZ3OHGCIJ/graph.json","events_json":"https://pith.science/api/pith-number/JAKQKRECMWMUDVXGSKZ3OHGCIJ/events.json","paper":"https://pith.science/paper/JAKQKREC"},"agent_actions":{"view_html":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ","download_json":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ.json","view_paper":"https://pith.science/paper/JAKQKREC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.14628&json=true","fetch_graph":"https://pith.science/api/pith-number/JAKQKRECMWMUDVXGSKZ3OHGCIJ/graph.json","fetch_events":"https://pith.science/api/pith-number/JAKQKRECMWMUDVXGSKZ3OHGCIJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/action/storage_attestation","attest_author":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/action/author_attestation","sign_citation":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/action/citation_signature","submit_replication":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/action/replication_record"}},"created_at":"2026-07-05T09:09:21.892647+00:00","updated_at":"2026-07-05T09:09:21.892647+00:00"}