{"paper":{"title":"The Additive Structure of Cartesian Products Spanning Few Distinct Distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.CO","authors_text":"Brandon Hanson","submitted_at":"2016-07-12T17:28:04Z","abstract_excerpt":"Guth and Katz proved that any point set $\\mathcal P$ in the plane determines $\\Omega(|\\mathcal P|/\\log|\\mathcal P|)$ distinct distances. We show that when near to this lower bound, a point set $\\mathcal P$ of the form $A\\times A$ must satisfy $|A-A|\\ll |A|^{2-1/8}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03442","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"}