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We show that the pairs of points in $P$ determine $\\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant.\n  We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over $\\F_p$ ($n<\\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute constant."},"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":"1001.1980","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-01-12T19:58:17Z","cross_cats_sorted":[],"title_canon_sha256":"fee055fd205d50db51f2c1d5fa535788f45f95cff72977cf24260f475cd63641","abstract_canon_sha256":"f09263753681f030555b247248aaf36445fc4e8627f78f928fbfcb4d408f5a64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:44.470006Z","signature_b64":"j2NhRcFiFeRjWu7gsxZA6TcUJAZ9iejC2dRTEbkZeL95KHLrX4phlbyb4SOhwm9eGLIq3PQo1L203WPEe0x9DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbc7b6fda19d3c31e37e8917ea9209396c28f4882bbed828f66937291e81f9b6","last_reissued_at":"2026-05-18T03:02:44.469244Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:44.469244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An explicit incidence theorem in F_p","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Harald Andres Helfgott, Misha Rudnev","submitted_at":"2010-01-12T19:58:17Z","abstract_excerpt":"Let $P = A\\times A \\subset \\mathbb{F}_p \\times \\mathbb{F}_p$, $p$ a prime. Assume that $P= A\\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\\mathbb{F}_p$. We show that the pairs of points in $P$ determine $\\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant.\n  We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over $\\F_p$ ($n<\\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute constant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.1980","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":"1001.1980","created_at":"2026-05-18T03:02:44.469372+00:00"},{"alias_kind":"arxiv_version","alias_value":"1001.1980v2","created_at":"2026-05-18T03:02:44.469372+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.1980","created_at":"2026-05-18T03:02:44.469372+00:00"},{"alias_kind":"pith_short_12","alias_value":"XPD3N7NBTU6D","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"XPD3N7NBTU6DDY36","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"XPD3N7NB","created_at":"2026-05-18T12:26:17.028572+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/XPD3N7NBTU6DDY36REL6VEQJHF","json":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF.json","graph_json":"https://pith.science/api/pith-number/XPD3N7NBTU6DDY36REL6VEQJHF/graph.json","events_json":"https://pith.science/api/pith-number/XPD3N7NBTU6DDY36REL6VEQJHF/events.json","paper":"https://pith.science/paper/XPD3N7NB"},"agent_actions":{"view_html":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF","download_json":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF.json","view_paper":"https://pith.science/paper/XPD3N7NB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1001.1980&json=true","fetch_graph":"https://pith.science/api/pith-number/XPD3N7NBTU6DDY36REL6VEQJHF/graph.json","fetch_events":"https://pith.science/api/pith-number/XPD3N7NBTU6DDY36REL6VEQJHF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF/action/storage_attestation","attest_author":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF/action/author_attestation","sign_citation":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF/action/citation_signature","submit_replication":"https://pith.science/pith/XPD3N7NBTU6DDY36REL6VEQJHF/action/replication_record"}},"created_at":"2026-05-18T03:02:44.469372+00:00","updated_at":"2026-05-18T03:02:44.469372+00:00"}