A new complexity function, repetitions in sturmian words, and irrationality exponents of sturmian numbers

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 ≥ 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,…,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 + 1a) for any integer a ≥ 34.