{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:XE2BHQM4MQYWRO74D7PCPB4XQZ","short_pith_number":"pith:XE2BHQM4","schema_version":"1.0","canonical_sha256":"b93413c19c643168bbfc1fde278797865e1d05e0480a744317bd734e1828a19c","source":{"kind":"arxiv","id":"1610.05620","version":2},"attestation_state":"computed","paper":{"title":"Collinear triples and quadruples for Cartesian products in $\\mathbb{F}_p^2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Giorgis Petridis","submitted_at":"2016-10-18T13:57:39Z","abstract_excerpt":"In this ote, which has been absorbed by arXiv1702.01003, we combine a recent point-line incidence bound of Stevens and de Zeeuw with an older lemma of Bourgain, Katz and Tao to bound the number of collinear triples and quadruples in a Cartesian product in $\\mathbb{F}_p^2$."},"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":"1610.05620","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-10-18T13:57:39Z","cross_cats_sorted":[],"title_canon_sha256":"492e40a7686fc68322b00cd99835086c8a22fe408f32012a75be125efe3f0a4e","abstract_canon_sha256":"dbd1d51d179dd54328e5d5c60af98c52ed0ca6e3b3a90bb075e77c1b3e2e9a71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:02.949733Z","signature_b64":"GOw2uhu8GVh0CEqRgrgGTE88mANSmwUYX0biDlLUeMYF7AV03KaQIZ5rqbaa30+J5Zxo5psleZ0Ea4zB/jofBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b93413c19c643168bbfc1fde278797865e1d05e0480a744317bd734e1828a19c","last_reissued_at":"2026-05-18T00:51:02.949201Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:02.949201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Collinear triples and quadruples for Cartesian products in $\\mathbb{F}_p^2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Giorgis Petridis","submitted_at":"2016-10-18T13:57:39Z","abstract_excerpt":"In this ote, which has been absorbed by arXiv1702.01003, we combine a recent point-line incidence bound of Stevens and de Zeeuw with an older lemma of Bourgain, Katz and Tao to bound the number of collinear triples and quadruples in a Cartesian product in $\\mathbb{F}_p^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05620","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":"1610.05620","created_at":"2026-05-18T00:51:02.949270+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.05620v2","created_at":"2026-05-18T00:51:02.949270+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.05620","created_at":"2026-05-18T00:51:02.949270+00:00"},{"alias_kind":"pith_short_12","alias_value":"XE2BHQM4MQYW","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"XE2BHQM4MQYWRO74","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"XE2BHQM4","created_at":"2026-05-18T12:30:51.357362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1907.03357","citing_title":"Some remarks on products of sets in the Heisenberg group and in the affine group","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ","json":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ.json","graph_json":"https://pith.science/api/pith-number/XE2BHQM4MQYWRO74D7PCPB4XQZ/graph.json","events_json":"https://pith.science/api/pith-number/XE2BHQM4MQYWRO74D7PCPB4XQZ/events.json","paper":"https://pith.science/paper/XE2BHQM4"},"agent_actions":{"view_html":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ","download_json":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ.json","view_paper":"https://pith.science/paper/XE2BHQM4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.05620&json=true","fetch_graph":"https://pith.science/api/pith-number/XE2BHQM4MQYWRO74D7PCPB4XQZ/graph.json","fetch_events":"https://pith.science/api/pith-number/XE2BHQM4MQYWRO74D7PCPB4XQZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ/action/storage_attestation","attest_author":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ/action/author_attestation","sign_citation":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ/action/citation_signature","submit_replication":"https://pith.science/pith/XE2BHQM4MQYWRO74D7PCPB4XQZ/action/replication_record"}},"created_at":"2026-05-18T00:51:02.949270+00:00","updated_at":"2026-05-18T00:51:02.949270+00:00"}