A Sturmian word of slope $q$ is the cutting sequence of a half-line $y=qx$. We establish a bijection between sequences of certain prefixes of the Sturmian word of slope $q$, and the $q$-decreasing words, which are binary words whose maximal factors of the form $0^a1^b$ satisfy $q \cdot a > b$ whenever $a>0$. We also show that the number of $q$-decreasing words of length $n$ grows as $\Phi(q)^{n(1 + o(1))}$, where $\Phi(1)$ is the golden ratio, $\Phi(2)$ is equal to the tribonacci constant, and that the function $\Phi(q)$ is strictly increasing, discontinuous at every rational point, and exhibits a nice fractal structure related to the Stern--Brocot tree and Minkowski's question mark function.

翻译：斜率为 $q$ 的 Sturmian 词是半直线 $y=qx$ 的切割序列。我们建立了斜率为 $q$ 的 Sturmian 词的若干特定前缀序列与 $q$-递减词之间的双射，其中 $q$-递减词是满足以下条件的二元词：其形如 $0^a1^b$ 的最大因子在 $a>0$ 时满足 $q \cdot a > b$。我们还证明了，长度为 $n$ 的 $q$-递减词的数量以 $\Phi(q)^{n(1 + o(1))}$ 的速度增长，其中 $\Phi(1)$ 是黄金比例，$\Phi(2)$ 等于 tribonacci 常数，并且函数 $\Phi(q)$ 严格递增，在每一个有理点处不连续，同时呈现出与 Stern–Brocot 树及 Minkowski 问号函数相关的优美分形结构。