{"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"}