A gap of the exponents of repetitions of Sturmian words

Abstract

By measuring second occurring times of factors of an infinite word $x$, Bugeaud and Kim introduced a new quantity $rep\left(x\right)$ called the exponent of repetition of $x$. It was proved by Bugeaud and Kim that $1\le rep\left(x\right)\le r_{max}=\sqrt{10}-\frac{3}{2}$ if $x$ is a Sturmian word. We determine the value $r_1$ such that there is no Sturmian word $x$ satisfying and $r_1$ is an accumulation point of the set of $rep\left(x\right)$ when $x$ runs over the Sturmian words.