A gap of the exponents of repetitions of Sturmian words

摘要

By measuring second occurring times of factors of an infinite word $x$, Bugeaud and Kim introduced a new quantity $operatorname{rep}(x)$ called the exponent of repetition of $x$. It was proved by Bugeaud and Kim that $1 leq operatorname{rep}(x) leq r_{max} = sqrt{10} - frac32$ if $x$ is a Sturmian word. We determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1 operatorname{rep}(x) r_{max}$ and $r_1$ is an accumulation point of the set of $operatorname{rep}(x)$ when $x$ runs over the Sturmian words.