{"paper":{"title":"Product mixing in the alternating group","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Sean Eberhard","submitted_at":"2015-12-11T04:29:01Z","abstract_excerpt":"We prove the following one-sided product-mixing theorem for the alternating group: Given subsets $X,Y,Z \\subset A_n$ of densities $\\alpha,\\beta,\\gamma$ satisfying $\\min(\\alpha\\beta,\\alpha\\gamma,\\beta\\gamma)\\gg n^{-1}(\\log n)^7$, there are at least $ (1+o(1))\\alpha\\beta\\gamma |A_n|^2$ solutions to $xy=z$ with $x\\in X, y\\in Y, z\\in Z$. One consequence is that the largest product-free subset of $A_n$ has density at most $n^{-1/2}(\\log n)^{7/2}$, which is best possible up to logarithms and improves the best previous bound of $n^{-1/3}$ due to Gowers. The main tools are a Fourier-analytic reduction"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03517","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"}