Anti-powers in infinite words

Gabriele Fici (1) , Antonio Restivo (2) , Manuel Silva , Luca Q. Zamboni (3, 4)

1 Università degli studi di Palermo - University of Palermo
2 DICGIM - Dipartimento di Ingegneria Chimica, Gestionale, Informatica, Meccanica [Palermo]
3 ICJ - Institut Camille Jordan
4 CTN - Combinatoire, théorie des nombres

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.

Mots clés: Unavoidable regularity; Infinite word; Anti-power

Domaines: Combinatoire [math.CO]

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