A gap of the exponents of repetitions of Sturmian words

By measuring the smallest second occuring time of every factor of an infinite word $x$, Bugeaud and Kim introduced a new quantity ${\rm rep}(x)$ called the exponent of repetition of $x$. Among other results, Bugeaud and Kim proved that $1 \leq {\rm rep}(x) \leq r_{\max} = \sqrt{10} - 3/2$ and $r_{\max}$ is the isolated maximum value when $x$ varies over the Sturmian words. In this paper, we determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1 < {\rm rep}(x) < r_{\max}$ and $r_1$ is an accumulate point of the set of ${\rm rep}(x)$ when $x$ runs over the Sturmian words.