{"paper":{"title":"On an application of Guth-Katz theorem","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, Oliver Roche-Newton","submitted_at":"2011-03-07T19:48:48Z","abstract_excerpt":"We prove that for some universal $c$, a non-collinear set of $N>\\frac{1}{c}$ points in the Euclidean plane determines at least $c \\frac{N}{\\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \\frac{N}{\\log N}$ distinct dot products.\n  This in particular implies a sum-product bound $$ |A\\cdot A\\pm A\\cdot A|\\geq c\\frac{|A|^2}{\\log |A|} $$ for a discrete $A \\subset {\\mathbb R}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1354","kind":"arxiv","version":5},"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"}