Asymptotic repetitive threshold of balanced sequences

Abstract

The critical exponent $E(\mathbf{u})$ of an infinite sequence $\mathbf{u}$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf{u}$. By the famous Dejean's theorem, $E(\mathbf{u})\ge 1+\frac{1}{d-1}$ for every $d$-ary sequence $\mathbf{u}$. We define the asymptotic critical exponent $E^{\ast }(\mathbf{u})$ as the upper limit of the maximal repetition of factors of length $n$. We show that for any $d>1$ there exists a $d$-ary sequence $\mathbf{u}$ having $E^{\ast }(\mathbf{u})$ arbitrarily close to $1$. Then we focus on the class of $d$-ary balanced sequences. In this class, the values $E^{\ast }(\mathbf{u})$ are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a $d$-ary balanced sequence with the least asymptotic critical exponent for $2\le d\le 10$.