{"paper":{"title":"Further improvements to incidence and Beck-type bounds over prime finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Timothy G. F. Jones","submitted_at":"2012-06-20T14:51:33Z","abstract_excerpt":"We establish improved finite field Szemeredi-Trotter and Beck type theorems. First we show that if P and L are a set of points and lines respectively in the plane F_p^2, with |P|,|L| \\leq N and N<p, then there are at most C_1 N^{3/2-1/662+o(1)} incidences between points in P and lines in L. Here C_1 is some absolute constant greater than 1. This improves on the previously best-known bound of C_1 N^{3/2-1/806+o(1)}.\n  Second we show that if P is a set of points in \\mathbb{F}_p^2 with |P|<p then either at least C_2|P|^{1-o(1)} points in P are contained in a single line, or P determines least C_2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4517","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"}