Anti-Powers in Infinite Words

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

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Gabriele Fici
Antonio Restivo
Manuel Silva
Luca Q. Zamboni

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Gabriele Fici, Antonio Restivo, Manuel Silva, and Luca Q. Zamboni. Anti-Powers in Infinite Words. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 124:1-124:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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.
  • infinite word
  • anti-power
  • unavoidable regularity
  • avoidability


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  1. J.-P. Allouche and J. Shallit. The ubiquitous prouhet-thue-morse sequence. In Sequences and their Applications: Proceedings of SETA'98, pages 1-16. Springer, 1999. Google Scholar
  2. A. Condon, J. Maňuch, and C. Thachuk. The complexity of string partitioning. J. Discrete Algorithms, 32(C):24-43, May 2015. URL:
  3. A. de Luca and S. Varricchio. Finiteness and Regularity in Semigroups and Formal Languages. Monographs in Theoretical Computer Science (An EATCS Series). Springer, 1st edition, 1999. Google Scholar
  4. H. Fernau, F. Manea, R. Mercaş, and M. L. Schmid. Pattern Matching with Variables: Fast Algorithms and New Hardness Results. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 302-315, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  5. M. Lothaire. Combinatorics on Words, 2nd edition. Cambridge Mathematical Library, Cambridge University Press, 1997. Google Scholar
  6. M. Lothaire. Algebraic Combinatorics on Words. Cambridge University Press, 2002. Google Scholar
  7. M. L. Schmid. Computing equality-free and repetitive string factorisations. Theoret. Comput. Sci., 618:42-51, March 2016. URL:
  8. A. Thue. Uber unendliche zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., pages 1-22, 1906. Google Scholar
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