{"paper":{"title":"Products of Differences over Arbitrary Finite Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Brendan Murphy, Giorgis Petridis","submitted_at":"2017-05-18T13:20:26Z","abstract_excerpt":"There exists an absolute constant $\\delta > 0$ such that for all $q$ and all subsets $A \\subseteq \\mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \\delta}$, then \\[ |(A-A)(A-A)| = |\\{ (a -b) (c-d) : a,b,c,d \\in A\\}| > \\frac{q}{2}. \\] Any $\\delta < 1/13,542$ suffices for sufficiently large $q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions.\n  Our proof is based on a qualitatively optimal characterisation of sets $A,X \\subseteq \\mathbb{F}_q$ for which the number of solutions to the e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.06581","kind":"arxiv","version":3},"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"}