{"paper":{"title":"Stronger sum-product inequalities for small sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"George Shakan, Ilya Shkredov, Misha Rudnev","submitted_at":"2018-08-25T20:03:17Z","abstract_excerpt":"Let $F$ be a field and a finite $A\\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$.\n  We strengthen the \"threshold\" sum-product inequality $$|AA|^3 |A\\pm A|^2 \\gg |A|^6\\,,\\;\\;\\;\\;\\mbox{hence} \\;\\; \\;\\;|AA|+|A+A|\\gg |A|^{1+\\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to\n  $$|AA|^5 |A\\pm A|^4 \\gg |A|^{11-o(1)}\\,,\\;\\;\\;\\;\\mbox{hence} \\;\\; \\;\\;|AA|+|A\\pm A|\\gg |A|^{1+\\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \\gg |A|^{73-o(1)}. $$ The latter inequality is \"threshold-breaking\", for it shows for $\\epsilon>0$, one has $$|AA| \\le |A|^{1+\\epsi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08465","kind":"arxiv","version":4},"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"}