Anti-Powers in Infinite Words
Gabriele Fici, Antonio Restivo, Manuel Silva, Luca Q. Zamboni

TL;DR
This paper explores the unavoidable presence of powers and anti-powers in infinite words, establishing conditions under which they must occur and characterizing words that avoid certain anti-power orders.
Contribution
It introduces the concept of anti-powers in infinite words and proves their unavoidable occurrence, also analyzing words that avoid anti-powers of specific orders.
Findings
Every infinite word contains powers or anti-powers of any order.
Anti-powers of every order start at every position in aperiodic uniformly recurrent words.
Words avoiding anti-powers of order 3 are ultimately periodic, while those avoiding order 4 or 6 can be aperiodic.
Abstract
In combinatorics of words, a concatenation of consecutive equal blocks is called a power of order . In this paper we take a different point of view and define an anti-power of order as a concatenation of 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 is ultimately periodic, while there exist aperiodic words avoiding anti-powers of order…
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