Abelian Repetition Threshold Revisited

Abstract

In combinatorics on words, repetition thresholds are the numbers separating avoidable and unavoidable repetitions of a given type in a given class of words. For example, the meaning of the “classical” repetition threshold $\mathsf{RT}(k)$ is “every infinite k-ary word contains an $\alpha$-power of a nonempty word for some $\alpha \ge \mathsf{RT}(k)$ but some infinite k-ary words contain no such $\alpha$-powers with $\alpha > \mathsf{RT}(k)$”. It is well known that $\mathsf{RT}(k)=\frac{k}{k-1}$ with the exceptions for $k=3,4$.

For Abelian repetition threshold $\mathsf{ART}(k)$, avoidance of fractional Abelian powers of words is considered. The exact values of $\mathsf{ART}(k)$ are unknown; the lower bound $\mathsf{ART}(2)\ge \frac{11}{3}$, $\mathsf{ART}(3)\ge 2$, $\mathsf{ART}(4)\ge \frac{9}{5}$, $\mathsf{ART}(k)\ge \frac{k-2}{k-3}$ for all $k\ge 5$ was proved by Samsonov and Shur in 2012 and conjectured to be tight. We present a method of study of Abelian power-free languages using random walks in prefix trees and some experimental results obtained by this method. On the base of these results, we suggest that the lower bounds for $\mathsf{ART}(k)$ by Samsonov and Shur are not tight for all k except $k= 5$. We prove $\mathsf{ART}(k)>\frac{k-2}{k-3}$ for $k=6,7,8,9,10$ and state a new conjecture on the Abelian repetition threshold.

E. A. Petrova—Supported by the Ministry of Science and Higher Education of the Russian Federation, project FEUZ-2020-0016.

A. M. Shur—Supported by Ural Mathematical Center, project 075-02-2022-877.