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Anti-powers in infinite words

Abstract : In combinatorics of words, a concatenation of k consecutive equal blocks is called a power of order k. In this paper we take a different point of view and define an anti-power of order k as a concatenation of k consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. As a consequence, we show that in every aperiodic uniformly recurrent word, anti-powers of every order begin at every position. We further show that every infinite word avoiding anti-powers of order 3 is ultimately periodic, while there exist aperiodic words avoiding anti-powers of order 4. We also show that there exist aperiodic recurrent words avoiding anti-powers of order 6.
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Submitted on : Tuesday, March 19, 2019 - 2:48:55 PM
Last modification on : Monday, June 28, 2021 - 2:26:07 PM
Long-term archiving on: : Thursday, June 20, 2019 - 12:27:51 PM

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Gabriele Fici, Antonio Restivo, Manuel Silva, Luca Q. Zamboni. Anti-powers in infinite words. Journal of Combinatorial Theory, Series A, Elsevier, 2018, 157, pp.109 - 119. ⟨10.1016/j.jcta.2018.02.009⟩. ⟨hal-01829146⟩

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