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 rep(x) called the exponent of repetition of x. It was proved by Bugeaud and Kim that 1 ≤ rep(x) ≤ r_(max) =√10-3/2 if x is a Sturmian word. We determine the value r_1 such that there is no Sturmian word x satisfying r_1 < rep(x) < rmax and r_1 is an accumulation point of the set of rep(x) when x runs over the Sturmian words.