Anti-powers in infinite words

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

Research output: Chapter in Book/Report/Conference proceedingChapter

5 Citations (Scopus)

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. From these results, we derive that at every position of an aperiodic uniformly recurrent word start anti-powers of any order. We further show that any infinite word avoiding anti-powers of order 3 is ultimately periodic, and that 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, and leave open the question whether there exist aperiodic recurrent words avoiding anti-powers of order k for k = 4, 5.
Original languageEnglish
Title of host publicationLeibniz International Proceedings in Informatics, LIPIcs
Number of pages9
Publication statusPublished - 2016

All Science Journal Classification (ASJC) codes

  • Software

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