{"paper":{"title":"Boolean functions on $S_n$ which are nearly linear","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Yuval Filmus","submitted_at":"2021-07-16T11:42:49Z","abstract_excerpt":"We show that if $f\\colon S_n \\to \\{0,1\\}$ is $\\epsilon$-close to linear in $L_2$ and $\\mathbb{E}[f] \\leq 1/2$ then $f$ is $O(\\epsilon)$-close to a union of \"mostly disjoint\" cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut-Kalai-Naor theorem for the symmetric group.\n  Using similar techniques, we show that if $f\\colon S_n \\to \\mathbb{R}$ is linear, $\\Pr[f \\notin \\{0,1\\}] \\leq \\epsilon$, and $\\Pr[f = 1] \\leq 1/2$, then $f$ is $O(\\epsilon)$-close to a union of mostly disjoint cosets, and this is also sharp; and that if $f\\colon S_n \\to \\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.07833","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2107.07833/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}