{"paper":{"title":"Friedgut--Kalai--Naor theorem for slices of the Boolean cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yuval Filmus","submitted_at":"2014-10-28T22:34:59Z","abstract_excerpt":"The Friedgut--Kalai--Naor theorem states that if a Boolean function $f\\colon \\{0,1\\}^n \\to \\{0,1\\}$ is close (in $L^2$-distance) to an affine function $\\ell(x_1,...,x_n) = c_0 + \\sum_i c_i x_i$, then $f$ is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions defined over $\\binom{[n]}{k} = \\{(x_1,...,x_n) \\in \\{0,1\\}^n : \\sum_i x_i = k \\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7834","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"}