{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:OO7XAJ2LNZNM4UPIO6JCZ7KEHH","short_pith_number":"pith:OO7XAJ2L","schema_version":"1.0","canonical_sha256":"73bf70274b6e5ace51e877922cfd4439ce3190ded6fb320261e9b39ee3b9ece7","source":{"kind":"arxiv","id":"1504.06989","version":1},"attestation_state":"computed","paper":{"title":"On the number of unit-area triangles spanned by convex grids in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilya D. Shkredov, Micha Sharir, Orit E. Raz","submitted_at":"2015-04-27T09:25:59Z","abstract_excerpt":"A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A, B\\subset\\mathbb R$, each of size $n^{1/2}$, the convex grid $A\\times B$ spans at most $O(n^{37/17}\\log^{2/17}n)$ unit-area triangles. This improves the best known upper bound $O(n^{31/14})$ recently obtained in \\cite{RS}. Our analysis also applies to more general families of sets $A$, $B$, known as sets of Szemer\\'edi--Trotter type."},"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":"1504.06989","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-27T09:25:59Z","cross_cats_sorted":[],"title_canon_sha256":"a3d975ce476513708cec3a89655ca807ace161619d73d3e6530bfa32136155a1","abstract_canon_sha256":"a3469c0c7e44ab89c662b6853a21146cea76984cdf4ed69026d93422d975770a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:43.956663Z","signature_b64":"6EFkBGsf35iFeCohBh/3e6pWnvX7umiuD7JWwb6JtCsPEEfV7wq9ok3jprzFiRkDHPqrxUO7iCTGW3liFrZoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73bf70274b6e5ace51e877922cfd4439ce3190ded6fb320261e9b39ee3b9ece7","last_reissued_at":"2026-05-18T02:17:43.956292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:43.956292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of unit-area triangles spanned by convex grids in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilya D. Shkredov, Micha Sharir, Orit E. Raz","submitted_at":"2015-04-27T09:25:59Z","abstract_excerpt":"A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A, B\\subset\\mathbb R$, each of size $n^{1/2}$, the convex grid $A\\times B$ spans at most $O(n^{37/17}\\log^{2/17}n)$ unit-area triangles. This improves the best known upper bound $O(n^{31/14})$ recently obtained in \\cite{RS}. Our analysis also applies to more general families of sets $A$, $B$, known as sets of Szemer\\'edi--Trotter type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06989","kind":"arxiv","version":1},"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":"1504.06989","created_at":"2026-05-18T02:17:43.956367+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.06989v1","created_at":"2026-05-18T02:17:43.956367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06989","created_at":"2026-05-18T02:17:43.956367+00:00"},{"alias_kind":"pith_short_12","alias_value":"OO7XAJ2LNZNM","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"OO7XAJ2LNZNM4UPI","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"OO7XAJ2L","created_at":"2026-05-18T12:29:34.919912+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/OO7XAJ2LNZNM4UPIO6JCZ7KEHH","json":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH.json","graph_json":"https://pith.science/api/pith-number/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/graph.json","events_json":"https://pith.science/api/pith-number/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/events.json","paper":"https://pith.science/paper/OO7XAJ2L"},"agent_actions":{"view_html":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH","download_json":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH.json","view_paper":"https://pith.science/paper/OO7XAJ2L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.06989&json=true","fetch_graph":"https://pith.science/api/pith-number/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/graph.json","fetch_events":"https://pith.science/api/pith-number/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/action/storage_attestation","attest_author":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/action/author_attestation","sign_citation":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/action/citation_signature","submit_replication":"https://pith.science/pith/OO7XAJ2LNZNM4UPIO6JCZ7KEHH/action/replication_record"}},"created_at":"2026-05-18T02:17:43.956367+00:00","updated_at":"2026-05-18T02:17:43.956367+00:00"}