Anti-powers in infinite words

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 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 publication43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
EditorsYuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Volume55
ISBN (Electronic)9783959770132
DOIs
Publication statusPublished - 1 Aug 2016
Event43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy
Duration: 12 Jul 201615 Jul 2016

Conference

Conference43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
Country/TerritoryItaly
CityRome
Period12/07/1615/07/16

Keywords

  • Anti-power
  • Avoidability
  • Infinite word
  • Unavoidable regularity

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