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We prove a similar theorem for functions defined over $\\binom{[n]}{k} = \\{(x_1,...,x_n) \\in \\{0,1\\}^n : \\sum_i x_i = k \\}$."},"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":"1410.7834","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-28T22:34:59Z","cross_cats_sorted":[],"title_canon_sha256":"01e9b932ef780c54e6147961ccd2aea8374fa5596b33902b9c7b245a15566b10","abstract_canon_sha256":"aa9ef384c7cc4fe88ffeef25124fe2bcdef6a261c5ac7e427013cb2dec78ad61"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:59.008400Z","signature_b64":"SSpbzA9CorXWsRrS86jUoc51GC7qPFTs/UrOIq+rqYJVQhVmKKWLdrpMBM7iViqi/ITpIC0jiPfo6nqQ4qGsCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2da2eb4f84987b43e6b41ea1528c85c11673ac888e7b0958a2025b906543eb9","last_reissued_at":"2026-05-18T01:15:59.007789Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:59.007789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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). 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