{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:2OTVJTGBFBXMJ6EZ7LYEJNLT7A","short_pith_number":"pith:2OTVJTGB","schema_version":"1.0","canonical_sha256":"d3a754ccc1286ec4f899faf044b573f818d2aeaf0d1efb67bd61557a4e483903","source":{"kind":"arxiv","id":"1501.00379","version":2},"attestation_state":"computed","paper":{"title":"The number of unit-area triangles in the plane: Theme and variations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM","math.MG"],"primary_cat":"math.CO","authors_text":"Micha Sharir, Orit E. Raz","submitted_at":"2015-01-02T11:02:12Z","abstract_excerpt":"We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if $S$ consists of points on three lines, the number of unit-area triangles that $S$ spans can be $\\Omega(n^2)$, for any triple of lines (it is always $O(n^2)$ in this case). (ii) We show that if $S$ is a {\\em convex grid} of the form $A\\times B$, where $A$, $B$ are {\\em convex}"},"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":"1501.00379","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-02T11:02:12Z","cross_cats_sorted":["cs.CG","cs.DM","math.MG"],"title_canon_sha256":"dfeefdd55f94c08488cccbba38ebc0017d9ffc6fd6dac040c8da3092b1ec4cc6","abstract_canon_sha256":"417375ff2c0af97d737e1a537052ba42bbe9c8920dbf576415059db1d9c69150"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:00.553994Z","signature_b64":"5l8+39RaW28gS9Lr6kWvpas0NmsF2qFOnVh6/TqoIr3IRKnUP5OC0zdsMgLDNBinPPMox9y6kW+EMG3dBdmHCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3a754ccc1286ec4f899faf044b573f818d2aeaf0d1efb67bd61557a4e483903","last_reissued_at":"2026-05-18T02:19:00.550306Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:00.550306Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of unit-area triangles in the plane: Theme and variations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM","math.MG"],"primary_cat":"math.CO","authors_text":"Micha Sharir, Orit E. Raz","submitted_at":"2015-01-02T11:02:12Z","abstract_excerpt":"We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if $S$ consists of points on three lines, the number of unit-area triangles that $S$ spans can be $\\Omega(n^2)$, for any triple of lines (it is always $O(n^2)$ in this case). (ii) We show that if $S$ is a {\\em convex grid} of the form $A\\times B$, where $A$, $B$ are {\\em convex}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00379","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":"1501.00379","created_at":"2026-05-18T02:19:00.553338+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.00379v2","created_at":"2026-05-18T02:19:00.553338+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00379","created_at":"2026-05-18T02:19:00.553338+00:00"},{"alias_kind":"pith_short_12","alias_value":"2OTVJTGBFBXM","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"2OTVJTGBFBXMJ6EZ","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"2OTVJTGB","created_at":"2026-05-18T12:29:02.477457+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/2OTVJTGBFBXMJ6EZ7LYEJNLT7A","json":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A.json","graph_json":"https://pith.science/api/pith-number/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/graph.json","events_json":"https://pith.science/api/pith-number/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/events.json","paper":"https://pith.science/paper/2OTVJTGB"},"agent_actions":{"view_html":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A","download_json":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A.json","view_paper":"https://pith.science/paper/2OTVJTGB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.00379&json=true","fetch_graph":"https://pith.science/api/pith-number/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/graph.json","fetch_events":"https://pith.science/api/pith-number/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/action/storage_attestation","attest_author":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/action/author_attestation","sign_citation":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/action/citation_signature","submit_replication":"https://pith.science/pith/2OTVJTGBFBXMJ6EZ7LYEJNLT7A/action/replication_record"}},"created_at":"2026-05-18T02:19:00.553338+00:00","updated_at":"2026-05-18T02:19:00.553338+00:00"}