{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:7RLGXOBY5B2TVNLGU2UYDEQJGD","short_pith_number":"pith:7RLGXOBY","schema_version":"1.0","canonical_sha256":"fc566bb838e8753ab566a6a981920930d8ce0a320308bf45381dec512ca56d58","source":{"kind":"arxiv","id":"1211.4917","version":2},"attestation_state":"computed","paper":{"title":"On arithmetic progressions in A + B + C","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kevin Henriot","submitted_at":"2012-11-21T02:29:26Z","abstract_excerpt":"Our main result states that when A, B, C are subsets of Z/NZ of respective densities \\alpha,\\beta,\\gamma, the sumset A + B + C contains an arithmetic progression of length at least e^{c(\\log N)^c} for densities \\alpha > (\\log N)^{-2 + \\epsilon} and \\beta,\\gamma > e^{-c(\\log N)^c}, where c depends on \\epsilon. Previous results of this type required one set to have density at least (\\log N)^{-1 + o(1)}. Our argument relies on the method of Croot, Laba and Sisask to establish a similar estimate for the sumset A + B and on the recent advances on Roth's theorem by Sanders. We also obtain new estima"},"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":"1211.4917","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-21T02:29:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1695b25d302ecc4e98cd6a91fee89dc61abfd304374e96ee97baf658cea41c1e","abstract_canon_sha256":"41523db224f18e5dafe122831de103e3308ae024eeeaf4bfc76f1b77d108cd74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:58.296263Z","signature_b64":"X6ac4ZNFAi0Ip0LUEXJpVZ2YtTvyZOMa93cmh6YjsTpthpjC/pHBTW69J+UwaFXUjJvcj9B8lxXcV7vXgfj5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc566bb838e8753ab566a6a981920930d8ce0a320308bf45381dec512ca56d58","last_reissued_at":"2026-05-18T03:10:58.295613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:58.295613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On arithmetic progressions in A + B + C","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kevin Henriot","submitted_at":"2012-11-21T02:29:26Z","abstract_excerpt":"Our main result states that when A, B, C are subsets of Z/NZ of respective densities \\alpha,\\beta,\\gamma, the sumset A + B + C contains an arithmetic progression of length at least e^{c(\\log N)^c} for densities \\alpha > (\\log N)^{-2 + \\epsilon} and \\beta,\\gamma > e^{-c(\\log N)^c}, where c depends on \\epsilon. Previous results of this type required one set to have density at least (\\log N)^{-1 + o(1)}. Our argument relies on the method of Croot, Laba and Sisask to establish a similar estimate for the sumset A + B and on the recent advances on Roth's theorem by Sanders. We also obtain new estima"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4917","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":""},"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":"1211.4917","created_at":"2026-05-18T03:10:58.295707+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.4917v2","created_at":"2026-05-18T03:10:58.295707+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4917","created_at":"2026-05-18T03:10:58.295707+00:00"},{"alias_kind":"pith_short_12","alias_value":"7RLGXOBY5B2T","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"7RLGXOBY5B2TVNLG","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"7RLGXOBY","created_at":"2026-05-18T12:26:58.693483+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD","json":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD.json","graph_json":"https://pith.science/api/pith-number/7RLGXOBY5B2TVNLGU2UYDEQJGD/graph.json","events_json":"https://pith.science/api/pith-number/7RLGXOBY5B2TVNLGU2UYDEQJGD/events.json","paper":"https://pith.science/paper/7RLGXOBY"},"agent_actions":{"view_html":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD","download_json":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD.json","view_paper":"https://pith.science/paper/7RLGXOBY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.4917&json=true","fetch_graph":"https://pith.science/api/pith-number/7RLGXOBY5B2TVNLGU2UYDEQJGD/graph.json","fetch_events":"https://pith.science/api/pith-number/7RLGXOBY5B2TVNLGU2UYDEQJGD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD/action/storage_attestation","attest_author":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD/action/author_attestation","sign_citation":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD/action/citation_signature","submit_replication":"https://pith.science/pith/7RLGXOBY5B2TVNLGU2UYDEQJGD/action/replication_record"}},"created_at":"2026-05-18T03:10:58.295707+00:00","updated_at":"2026-05-18T03:10:58.295707+00:00"}