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.

Fici, G., Restivo, A., Silva, M., Zamboni, L.Q. (2018). Anti-powers in infinite words. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 157, 109-119 [10.1016/j.jcta.2018.02.009].

### Anti-powers in infinite words

#### 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. 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.
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2018
Fici, G., Restivo, A., Silva, M., Zamboni, L.Q. (2018). Anti-powers in infinite words. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 157, 109-119 [10.1016/j.jcta.2018.02.009].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10447/279725`