{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0107","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"}