{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T12:59:40Z","timestamp":1760101180593},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2019,10,18]],"date-time":"2019-10-18T00:00:00Z","timestamp":1571356800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2020,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Friedgut\u2013Kalai\u2013Naor (FKN) theorem states that if \u0192 is a Boolean function on the Boolean cube which is close to degree one, then \u0192 is close to a <jats:italic>dictator<\/jats:italic>, a function depending on a single coordinate. The author has extended the theorem to the <jats:italic>slice<\/jats:italic>, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the <jats:italic>multislice<\/jats:italic>, a multicoloured version of the slice.<\/jats:p><jats:p>As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.<\/jats:p>","DOI":"10.1017\/s0963548319000361","type":"journal-article","created":{"date-parts":[[2019,10,18]],"date-time":"2019-10-18T09:11:34Z","timestamp":1571389894000},"page":"200-212","source":"Crossref","is-referenced-by-count":2,"title":["FKN theorem for the multislice, with applications"],"prefix":"10.1017","volume":"29","author":[{"given":"Yuval","family":"Filmus","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,10,18]]},"reference":[{"key":"S0963548319000361_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139814782"},{"key":"S0963548319000361_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2018.11.006"},{"key":"S0963548319000361_ref6","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2011-00690-5"},{"key":"S0963548319000361_ref5","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20515"},{"key":"S0963548319000361_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-014-3027-1"},{"key":"S0963548319000361_ref12","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2015.v011a018"},{"key":"S0963548319000361_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-8858(02)00024-6"},{"key":"S0963548319000361_ref3","unstructured":"[3] Dikstein, Y. , Dinur, I. , Filmus, Y. and Harsha, P. (2018) Boolean function analysis on high-dimensional expanders. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX\/RANDOM 2018), Vol. 116 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl\u2013Leibniz-Zentrum f\u00fcr Informatik, pp. 38:1\u201338:20."},{"key":"S0963548319000361_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00535487"},{"key":"S0963548319000361_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-004-0478-3"},{"key":"S0963548319000361_ref7","unstructured":"[7] Filmus, Y. (2016) Friedgut\u2013Kalai\u2013Naor theorem for slices of the Boolean cube. Chic. J. Theoret. Comput. Sci. 2016 14:1\u201314:17."},{"key":"S0963548319000361_ref11","unstructured":"[11] Frobenius, G. (1900) \u00dcber die Charaktere der symmetrischen Gruppe. Sitzungsberichte der K\u00f6niglich preussischen Akademie der Wissenschaften zu Berlin, pp. 516\u2013534."},{"key":"S0963548319000361_ref9","unstructured":"[9] Filmus, Y. , O\u2019Donnell, R. and Wu, X. (2018) A log-Sobolev inequality for the multislice, with applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), Vol. 124 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl\u2013Leibniz-Zentrum f\u00fcr Informatik, pp. 34:1\u201334:12."},{"key":"S0963548319000361_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-6804-6"},{"key":"S0963548319000361_ref13","doi-asserted-by":"publisher","DOI":"10.4064\/cm137-2-9"},{"key":"S0963548319000361_ref15","unstructured":"[15] Rubinstein, A. (2012) Boolean functions whose Fourier transform is concentrated on pairwise disjoint subsets of the input. Master\u2019s thesis, Tel Aviv University, Israel."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548319000361","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,25]],"date-time":"2020-03-25T08:14:57Z","timestamp":1585124097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548319000361\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,18]]},"references-count":16,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["S0963548319000361"],"URL":"https:\/\/doi.org\/10.1017\/s0963548319000361","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,18]]}}}