A gap of the exponents of repetitions of Sturmian words

By measuring second occurring times of factors of an infinite word $x$, Bugeaud and Kim introduced a new quantity $\mathrm{r}\mathrm{e}\mathrm{p}(x)$ called the exponent of repetition of $x$. It was proved by Bugeaud and Kim that $1 \leq {\rm rep}(x) \leq r_{\max} = \sqrt{10} - 3/2$if$x$ is a Sturmian words. In this paper, we determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1<\mathrm{r}\mathrm{e}\mathrm{p}(x)<r_{\max }$ and $r_1$ is an accumulation point of the set of $\mathrm{r}\mathrm{e}\mathrm{p}(x)$ when $x$ runs over the Sturmian words.