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 positive 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词的某些前缀序列与$q$-递减词之间的双射，后者是一种二元词，其形如$0^a1^b$的最大因子满足当$a>0$时$q \cdot a > b$。我们还证明了长度为$n$的$q$-递减词的数量以$\Phi(q)^{n(1 + o(1))}$增长，其中$\Phi(1)$是黄金比例，$\Phi(2)$等于三倍纳西常数，并且函数$\Phi(q)$严格递增，在每个正有理点处不连续，展现出与Stern--Brocot树及Minkowski问号函数相关的优美分形结构。