{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:JAKQKRECMWMUDVXGSKZ3OHGCIJ","short_pith_number":"pith:JAKQKREC","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"},"canonical_sha256":"4815054482659941d6e692b3b71cc242451addcb6a7e9eb78e39371cf3d401f0","source":{"kind":"arxiv","id":"2404.14628","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.14628","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"arxiv_version","alias_value":"2404.14628v2","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.14628","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"pith_short_12","alias_value":"JAKQKRECMWMU","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"pith_short_16","alias_value":"JAKQKRECMWMUDVXG","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"pith_short_8","alias_value":"JAKQKREC","created_at":"2026-07-05T09:09:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:JAKQKRECMWMUDVXGSKZ3OHGCIJ","target":"record","payload":{"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"},"canonical_sha256":"4815054482659941d6e692b3b71cc242451addcb6a7e9eb78e39371cf3d401f0","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"},"source_kind":"arxiv","source_id":"2404.14628","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:09:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n+TwpAEQ7n0nDEBetY80poUsBtEB2g9iNuck0Rm1gY32QkQWRruANf4ZOeDnS5G2Ra8VWZV06rBQqbQOPSCWAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T22:47:52.935147Z"},"content_sha256":"9e7b1057c886e64af9303af556d94a028192aaa0628dc1b7687265708fcd5895","schema_version":"1.0","event_id":"sha256:9e7b1057c886e64af9303af556d94a028192aaa0628dc1b7687265708fcd5895"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:JAKQKRECMWMUDVXGSKZ3OHGCIJ","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:09:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K2rAyWoBkQELcAy5n1kKj4S6QUGrYUo3mjSk2rwy7e2+GVcTNG9XJ+ZXgwT6pFo/Kvo31DTX1V5iCKRM+9uaAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T22:47:52.935675Z"},"content_sha256":"92c493fefa2dc1c66ef8e476c7262a7fea1e153cc298113490ffbe4fe4ed93c7","schema_version":"1.0","event_id":"sha256:92c493fefa2dc1c66ef8e476c7262a7fea1e153cc298113490ffbe4fe4ed93c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/bundle.json","state_url":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-19T22:47:52Z","links":{"resolver":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ","bundle":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/bundle.json","state":"https://pith.science/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JAKQKRECMWMUDVXGSKZ3OHGCIJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:JAKQKRECMWMUDVXGSKZ3OHGCIJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7c84224ce6254119ccc2dae02681685f2b8dc69ac8c0d05ce7e2f9880369c368","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-22T23:54:58Z","title_canon_sha256":"5469190d2de83bded704a4e76556c790873ebfa85f56c9b4976c1fff4a8c38a8"},"schema_version":"1.0","source":{"id":"2404.14628","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.14628","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"arxiv_version","alias_value":"2404.14628v2","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.14628","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"pith_short_12","alias_value":"JAKQKRECMWMU","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"pith_short_16","alias_value":"JAKQKRECMWMUDVXG","created_at":"2026-07-05T09:09:21Z"},{"alias_kind":"pith_short_8","alias_value":"JAKQKREC","created_at":"2026-07-05T09:09:21Z"}],"graph_snapshots":[{"event_id":"sha256:92c493fefa2dc1c66ef8e476c7262a7fea1e153cc298113490ffbe4fe4ed93c7","target":"graph","created_at":"2026-07-05T09:09:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2404.14628/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Daodao Yang, Dimitris Koukoulopoulos, James Maynard","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-22T23:54:58Z","title":"An almost sharp quantitative version of the Duffin-Schaeffer conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.14628","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9e7b1057c886e64af9303af556d94a028192aaa0628dc1b7687265708fcd5895","target":"record","created_at":"2026-07-05T09:09:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7c84224ce6254119ccc2dae02681685f2b8dc69ac8c0d05ce7e2f9880369c368","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-22T23:54:58Z","title_canon_sha256":"5469190d2de83bded704a4e76556c790873ebfa85f56c9b4976c1fff4a8c38a8"},"schema_version":"1.0","source":{"id":"2404.14628","kind":"arxiv","version":2}},"canonical_sha256":"4815054482659941d6e692b3b71cc242451addcb6a7e9eb78e39371cf3d401f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4815054482659941d6e692b3b71cc242451addcb6a7e9eb78e39371cf3d401f0","first_computed_at":"2026-07-05T09:09:21.892564Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:09:21.892564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qgeYt5fneAqDlH+Tu0pKXoqpWR83MJq4Z0YUaaX30IiEuIF7hGMVATL001MBWitGhMRoivokGjIitJwu5zZfBw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:09:21.893163Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.14628","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9e7b1057c886e64af9303af556d94a028192aaa0628dc1b7687265708fcd5895","sha256:92c493fefa2dc1c66ef8e476c7262a7fea1e153cc298113490ffbe4fe4ed93c7"],"state_sha256":"a5b30c542f2fbfd2b95d75b657696edee6884965c7500e2c4a05775aced67f05"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JJ7XfT0/7w2jlMGVSFk2BmgJLpMpcFxWQKpe3YDQJntwzEEL5cg6GW6ZNfhtJ58B7DfBCtIsiW/DFYGlfgIkBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T22:47:52.940814Z","bundle_sha256":"0b725fa2ad3fcc11ea9172cb8e3522e43f812c94e8a12a62c2fd7c94e41e08b6"}}