{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:L55SDANWZXCHBTAYAS7YRMZXPH","short_pith_number":"pith:L55SDANW","schema_version":"1.0","canonical_sha256":"5f7b2181b6cdc470cc1804bf88b33779c0364b7c695ddc31abac8446a903ab07","source":{"kind":"arxiv","id":"1608.06398","version":1},"attestation_state":"computed","paper":{"title":"An improvement on the number of simplices in $\\mathbb{F}_q^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anh Vinh Le, Duc Hiep Pham, Thang Pham","submitted_at":"2016-08-23T06:46:38Z","abstract_excerpt":"Let $\\mathcal{E}$ be a set of points in $\\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\\mathcal{E}|\\gg q^{d-\\frac{d-1}{k+1}}$ then $\\mathcal{E}$ determines a positive proportion of all $k$-simplices. In this paper, we give an improvement of this result in the case when $\\mathcal{E}$ is the Cartesian product of sets. More precisely, we show that if $\\mathcal{E}$ is the Cartesian product of sets and $q^{\\frac{kd}{k+1-1/d}}=o(|\\mathcal{E}|)$, the number of congruence classes of $k$-simplices determined by $\\mathcal{E}$ is at least $(1-o(1))q^{\\binom{k+1"},"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":"1608.06398","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-23T06:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"b6b4f7c15e9e46cddf012a7f9167bab505bf3253a05db1cd12af6405c3292428","abstract_canon_sha256":"ebdfa74e6ebcb4298c2515b07da008d3817b88e5f1387e4f470341368069c244"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:14.142781Z","signature_b64":"JfNKR41sETNl25LyXTrxlP9jOq309Z8BEJ3kYeeZYNd18ZFpquBWp81HJeMoQKl4SP9oMnXH9rt9x8yOUkaECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f7b2181b6cdc470cc1804bf88b33779c0364b7c695ddc31abac8446a903ab07","last_reissued_at":"2026-05-18T01:08:14.142240Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:14.142240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An improvement on the number of simplices in $\\mathbb{F}_q^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anh Vinh Le, Duc Hiep Pham, Thang Pham","submitted_at":"2016-08-23T06:46:38Z","abstract_excerpt":"Let $\\mathcal{E}$ be a set of points in $\\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\\mathcal{E}|\\gg q^{d-\\frac{d-1}{k+1}}$ then $\\mathcal{E}$ determines a positive proportion of all $k$-simplices. In this paper, we give an improvement of this result in the case when $\\mathcal{E}$ is the Cartesian product of sets. More precisely, we show that if $\\mathcal{E}$ is the Cartesian product of sets and $q^{\\frac{kd}{k+1-1/d}}=o(|\\mathcal{E}|)$, the number of congruence classes of $k$-simplices determined by $\\mathcal{E}$ is at least $(1-o(1))q^{\\binom{k+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06398","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":"1608.06398","created_at":"2026-05-18T01:08:14.142330+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.06398v1","created_at":"2026-05-18T01:08:14.142330+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06398","created_at":"2026-05-18T01:08:14.142330+00:00"},{"alias_kind":"pith_short_12","alias_value":"L55SDANWZXCH","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"L55SDANWZXCHBTAY","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"L55SDANW","created_at":"2026-05-18T12:30:29.479603+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/L55SDANWZXCHBTAYAS7YRMZXPH","json":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH.json","graph_json":"https://pith.science/api/pith-number/L55SDANWZXCHBTAYAS7YRMZXPH/graph.json","events_json":"https://pith.science/api/pith-number/L55SDANWZXCHBTAYAS7YRMZXPH/events.json","paper":"https://pith.science/paper/L55SDANW"},"agent_actions":{"view_html":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH","download_json":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH.json","view_paper":"https://pith.science/paper/L55SDANW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.06398&json=true","fetch_graph":"https://pith.science/api/pith-number/L55SDANWZXCHBTAYAS7YRMZXPH/graph.json","fetch_events":"https://pith.science/api/pith-number/L55SDANWZXCHBTAYAS7YRMZXPH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH/action/storage_attestation","attest_author":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH/action/author_attestation","sign_citation":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH/action/citation_signature","submit_replication":"https://pith.science/pith/L55SDANWZXCHBTAYAS7YRMZXPH/action/replication_record"}},"created_at":"2026-05-18T01:08:14.142330+00:00","updated_at":"2026-05-18T01:08:14.142330+00:00"}