A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers
Yann Bugeaud, Dong Han Kim · arXiv (Cornell University) · 2015
We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let $b \\ge 2$ be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of $b$-ary digits is a Sturmian sequence over $\\{0,1,\\ldots, b-1\\}$ and we prove that this lower bound is best possible. As an application, we derive some information on the $b$-ary expansion of $\\log(1+\\frac{1}{a})$,for any integer $a \\ge 34$.