Anti-powers in infinite words

Gabriele Fici, Antonio Restivo, Manuel Silva, Luca Q. Zamboni

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)109-119
Number of pages11
JournalJournal of Combinatorial Theory. Series A
Publication statusPublished - 1 Jul 2018


  • Anti-power
  • Infinite word
  • Unavoidable regularity


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