Abelian Repetition Threshold Revisited

TL;DR

This paper investigates the Abelian repetition threshold ART(k), proposing a new method using random walks in prefix trees, providing evidence that the known lower bounds are not tight for most k values, and proving this for k=6 to 10.

Contribution

It introduces a novel approach to study Abelian power-free languages and refutes existing conjectures about ART(k) for most k values, except for k=5.

Findings

ART(k) > (k-2)/(k-3) for k=6 to 10

Lower bounds by Samsonov and Shur are not tight for these k

Conjecture confirmed for k=6,7,8,9,10

Abstract

Abelian repetition threshold ART(k) is the number separating fractional Abelian powers which are avoidable and unavoidable over the k-letter alphabet. The exact values of ART(k) are unknown; the lower bounds were proved in [A.V. Samsonov, A.M. Shur. On Abelian repetition threshold. RAIRO ITA, 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 conjecture that the lower bounds for ART(k) by Samsonov and Shur are not tight for all k except for k=5 and prove this conjecture for k=6,7,8,9,10. Namely, we show that ART(k) > (k-2)/(k-3) in all these cases.