Anti-Powers in Infinite Words

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 $k$ 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 $6$.