We study the avoidability of long \n\n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-abelian-squares and \n\n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-abelian-cubes on binary and ternary alphabets. For \n\n \n \n k<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n k=1<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, these are M\u00e4kel\u00e4\u2019s questions. We show that one cannot avoid abelian-cubes of abelian period at least \n\n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in infinite binary words, and therefore answering negatively one question from M\u00e4kel\u00e4. Then we show that one can avoid \n\n \n 3<\/mml:mn>\n 3<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-abelian-squares of period at least \n\n \n 3<\/mml:mn>\n 3<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in infinite binary words and \n\n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-abelian-squares of period at least 2 in infinite ternary words. Finally, we study the minimum number of distinct \n\n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-abelian-squares that must appear in an infinite binary word.<\/p>","DOI":"10.1090\/mcom\/3085","type":"journal-article","created":{"date-parts":[[2015,7,15]],"date-time":"2015-07-15T13:46:58Z","timestamp":1436968018000},"page":"3051-3060","source":"Crossref","is-referenced-by-count":7,"title":["Avoidability of long \ud835\udc58-abelian repetitions"],"prefix":"10.1090","volume":"85","author":[{"given":"Micha\u00ebl","family":"Rao","sequence":"first","affiliation":[]},{"given":"Matthieu","family":"Rosenfeld","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2016,2,18]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1142\/S0218196793000123","article-title":"On abelian power-free morphisms","volume":"3","author":"Carpi, Arturo","year":"1993","journal-title":"Internat. 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Nauk SSSR","ISSN":"http:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"8","first-page":"Research Paper 2, approx. 9","article-title":"How many squares must a binary sequence contain?","volume":"2","author":"Fraenkel, Aviezri S.","year":"1995","journal-title":"Electron. J. Combin."},{"issue":"8","key":"9","doi-asserted-by":"publisher","first-page":"2189","DOI":"10.1016\/j.jcta.2013.08.008","article-title":"On a generalization of Abelian equivalence and complexity of infinite words","volume":"120","author":"Karhumaki, Juhani","year":"2013","journal-title":"J. Combin. Theory Ser. 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