Abelian Repetition Threshold Revisited

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 = 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.

On minimal critical exponent of balanced sequences

“…Replacing equality in the notion of a power with Abelian equivalence, Abelian powers, Abelian exponents and Abelian critical exponents are defined. There exist sequences with Abelian critical exponent arbitrarily close to 1 [8], but for no alphabet size d the minimal Abelian critical exponent of a d-ary sequence is known; see [28,25] for the best known lower bounds.…”