For an infinite word $x$, Bugeaud and Kim introduced a quantity $\mathrm{rep}(x)$ called the exponent of repetition of $x$. We prove that $\mathrm{rep}(x) = \mathrm{rep}(y)$ holds for a Sturmian word $x$ and every suffix $y$ of $x$. Let $c$ be the characteristic Sturmian word of slope $θ$. When $θ$ is a quadratic irrational, a formula of $\mathrm{rep}(c)$ is given. This formula shows that $\mathrm{rep}(c) = \mathrm{rep}(c')$ if $c'$ is the characteristic Sturmian word whose slope is a quadratic irrational equivalent to $θ$.
对于无限字$\mathbf{x}$，Bugeaud和Kim引入了一个新的复杂度函数$\text{rep}(\mathbf{x})$，称为$\mathbf{x}$的重复指数。他们显示了任何Sturmian单词$\mathbf{x}$的$1\le \text{rep}(\mathbf{x}) \le \sqrt{10}-\frac{3}{2}$。Ohnaka和Watanabe在Sturmian词的重复指数集合中发现了一个缺口。对于无理数$θ\in(0,1)$，令\[ \mathscr{L}(θ):=\{\text{rep}(\mathbf{x}):\textrm{$\mathbf{x}$ is an Sturmian word of slope $θ$}\}.\]在本文中，我们研究$\mathscr{L}(θ)$。确定$\mathscr{L}(θ)$的最小值，其中$θ$在其连分数表达式中具有有界偏导数。特别地，我们找到了$\mathscr{L}(φ)$的最大值和最小值，其中$φ:=\frac{\sqrt{5}-1}{2}$是黄金比例的分数部分。此外，我们表明，三个最大的值是孤立点在$\mathscr{L}(φ)$和第四大点是一个极限点的$\mathscr{L}(φ)$。
Alternating geometric progressions modulo one and Sturmian words
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of Chinese Academy of Sciences（中国科学院大学）
