Anti-Powers in Infinite Words
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
124:1-124:9
Regular Paper
Gabriele
Fici
Gabriele Fici
Antonio
Restivo
Antonio Restivo
Manuel
Silva
Manuel Silva
Luca Q.
Zamboni
Luca Q. Zamboni
10.4230/LIPIcs.ICALP.2016.124
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