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Comp."],"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove Bogolyubov\u2013Ruzsa-type results for finite subsets of groups with small tripling, |<jats:italic>A<\/jats:italic><jats:sup>3<\/jats:sup>| \u2264 <jats:italic>O<\/jats:italic>(|<jats:italic>A<\/jats:italic>|), or small alternation, |<jats:italic>AA<\/jats:italic><jats:sup>\u22121<\/jats:sup><jats:italic>A<\/jats:italic>| \u2264 <jats:italic>O<\/jats:italic>(|<jats:italic>A<\/jats:italic>|). As applications, we obtain a qualitative analogue of Bogolyubov\u2019s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.<\/jats:p>","DOI":"10.1017\/s0963548320000176","type":"journal-article","created":{"date-parts":[[2020,6,30]],"date-time":"2020-06-30T09:09:51Z","timestamp":1593508191000},"page":"807-829","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["On finite sets of small tripling or small alternation in arbitrary groups"],"prefix":"10.1017","volume":"29","author":[{"given":"Gabriel","family":"Conant","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,6,30]]},"reference":[{"key":"S0963548320000176_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004118000798"},{"key":"S0963548320000176_ref5","first-page":"185","article-title":"Sur quelques propri\u00e9t\u00e9s arithm\u00e9tiques des presque-p\u00e9riodes","volume":"4","author":"Bogolio\u00f9boff","year":"1939","journal-title":"Ann. 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(1969) Eigenschaften und Absch\u00e4tzungen von Wirkungsfunktionen, BMwF-GMD-22, Gesellschaft f\u00fcr Mathematik und Datenverarbeitung."},{"key":"S0963548320000176_ref20","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-07-00558-9"},{"key":"S0963548320000176_ref32","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788710000236"},{"key":"S0963548320000176_ref4","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1997.7042"},{"key":"S0963548320000176_ref27","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061304000346"},{"key":"S0963548320000176_ref23","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-2013-05820-5"},{"key":"S0963548320000176_ref16","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdl021"},{"key":"S0963548320000176_ref19","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2011-00708-X"},{"key":"S0963548320000176_ref30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01876039"},{"key":"S0963548320000176_ref39","unstructured":"[39] Terry, C. and Wolf, J. (2018) Quantitative structure of stable sets in finite abelian groups. Trans. Amer. Math. Soc. accepted."},{"key":"S0963548320000176_ref31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2418-1_21"},{"key":"S0963548320000176_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0072977"},{"key":"S0963548320000176_ref24","doi-asserted-by":"publisher","DOI":"10.5802\/jep.17"},{"key":"S0963548320000176_ref38","doi-asserted-by":"publisher","DOI":"10.1112\/blms.12211"},{"key":"S0963548320000176_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-012-0043-9"},{"key":"S0963548320000176_ref25","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/275"},{"key":"S0963548320000176_ref33","doi-asserted-by":"publisher","DOI":"10.2140\/apde.2012.5.627"},{"key":"S0963548320000176_ref34","unstructured":"[34] Sisask, O. (2018) Convolutions of sets with bounded VC-dimension are uniformly continuous.arXiv:1802.02836"},{"key":"S0963548320000176_ref12","unstructured":"[12] Freman, G. A. (1973) Foundations of a Structural Theory of Set Addition, Vol. 37 of Translations of Mathematical Monographs, AMS."},{"key":"S0963548320000176_ref10","unstructured":"[10] Conant, G. , Pillay, A. and Terry, C. (2018) Structure and regularity for subsets of groups with finite VC-dimension.arXiv:1802.04246"},{"key":"S0963548320000176_ref26","doi-asserted-by":"publisher","DOI":"10.5802\/jep.69"},{"key":"S0963548320000176_ref28","doi-asserted-by":"publisher","DOI":"10.4064\/fm275-7-2016"},{"key":"S0963548320000176_ref35","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-008-2271-7"},{"key":"S0963548320000176_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2019.01.033"},{"key":"S0963548320000176_ref18","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2008.167.601"},{"key":"S0963548320000176_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(64)50021-9"},{"key":"S0963548320000176_ref3","unstructured":"[3] Alon, N. , Fox, J. and Zhao, Y. (2019) Efficient arithmetic regularity and removal lemmas for induced bipartite patterns. Discrete Anal., Paper no. 3, 14 MR3943117."},{"key":"S0963548320000176_ref1","first-page":"224","article-title":"On approximations of groups, group actions and Hopf algebras","volume":"256","author":"Alekseev","year":"1999","journal-title":"Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)"},{"key":"S0963548320000176_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-010-0101-8"},{"key":"S0963548320000176_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548307008826"},{"key":"S0963548320000176_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-005-0509-8"},{"key":"S0963548320000176_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s000390050105"},{"key":"S0963548320000176_ref36","unstructured":"[36] Tao, T. 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