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\n The critical exponent\n \n \n \n \n E<\/mml:mi>\n (<\/mml:mo>\n \n u<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n E(\\mathbf u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of an infinite sequence\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over a finite alphabet expresses the maximal repetition of a factor in\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . By the famous Dejean\u2019s theorem,\n \n \n \n \n E<\/mml:mi>\n (<\/mml:mo>\n \n u<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \n 1<\/mml:mn>\n \n d<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n E(\\mathbf u) \\geq 1+\\frac 1{d-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for every\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -ary sequence\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We define the asymptotic critical exponent\n \n \n \n \n \n E<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n u<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n E^*(\\mathbf u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as the upper limit of the maximal repetition of factors of length\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We show that for any\n \n \n \n \n d<\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d>1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n there exists a\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -ary sequence\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n having\n \n \n \n \n \n E<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n u<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n E^*(\\mathbf u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n arbitrarily close to\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Then we focus on the class of\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -ary balanced sequences. In this class, the values\n \n \n \n \n \n E<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n u<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n E^*(\\mathbf u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -ary balanced sequence with the least asymptotic critical exponent for\n \n \n \n \n 2<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n d<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 10<\/mml:mn>\n <\/mml:mrow>\n 2\\leq d\\leq 10<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/mcom\/3816","type":"journal-article","created":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T09:41:25Z","timestamp":1671615685000},"page":"1403-1429","source":"Crossref","is-referenced-by-count":3,"title":["Asymptotic repetitive threshold of balanced sequences"],"prefix":"10.1090","volume":"92","author":[{"given":"L\u2019ubom\u00edra","family":"Dvo\u0159\u00e1kov\u00e1","sequence":"first","affiliation":[]},{"given":"Daniela","family":"Opo\u010densk\u00e1","sequence":"additional","affiliation":[]},{"given":"Edita","family":"Pelantov\u00e1","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,1,11]]},"reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"37","DOI":"10.4204\/EPTCS.63.7","article-title":"Finite-repetition threshold for infinite ternary words","volume":"63","author":"Badkobeh, G.","year":"2011","journal-title":"Electron. 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