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The result follows from a finite field version of the Beck theorem for large subsets of $\\F_q^2$ that we prove. If $|E|\\geq 64q\\log_2 q$, there exists a point $z\\in E$, such that there are at least $\\frac{q}{4}$ straight lines incide"},"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":"1205.0107","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-01T08:13:30Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"e66717c6b3db3eabfa3bc3218cbb53a990a78604a20ba5a773c33a5fb52a7f28","abstract_canon_sha256":"2c3287f6c99b417d4cd5b5e0a08734149022a0305e4760d8cdac9e6ea2d602a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:35.854622Z","signature_b64":"7wewCChash0AvjVfNrCbiuhAIIkTAtYh1In8r8j1O1QzKiiC6ZZoEjI1Ipg8TT2w/SA2ZeJ2FadG7SwPk/+vBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf3ebb5f9619cc92c9c396d56407c1d7c7f1b0a492f96b4a22bfe8b009cfdb64","last_reissued_at":"2026-05-18T03:56:35.854149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:35.854149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Areas of triangles and Beck's theorem in planes over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Misha Rudnev, Yujia Zhai","submitted_at":"2012-05-01T08:13:30Z","abstract_excerpt":"It is shown that any subset $E$ of a plane over a finite field $\\F_q$, of cardinality $|E|>q$ determines not less than $\\frac{q-1}{2}$ distinct areas of triangles, moreover once can find such triangles sharing a common base.\n  It is also shown that if $|E|\\geq 64q\\log_2 q$, then there are more than $\\frac{q}{2}$ distinct areas of triangles sharing a common vertex. 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