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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.03517","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2015-12-11T04:29:01Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2553754b3eb66fce4994a5600254ad49976bccad2fd3786d5c2aa7fcc4778305","abstract_canon_sha256":"7dcf1621d059d0dc98eea7a631715e98e7a8a6d25f804b5fdf2847474255b823"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:57.556782Z","signature_b64":"9CvJdwQHIYJQ3XNYXVvLrZeowU7fpm3swICBjCiWw2Kc9RHdHOEJ0NnGnMP3Rb6/+9zGGAQhdS1KP4gKd5KTDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56fa13be600c4e24551321c4e06e25304601c82615c61fafa44f05637b6ead27","last_reissued_at":"2026-05-18T00:50:57.556187Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:57.556187Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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