A NEW COMPLEXITY FUNCTION, REPETITIONS IN STURMIAN WORDS, AND IRRATIONALITY EXPONENTS OF STURMIAN NUMBERS

Abstract

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 + 1/a for any integer a >= 34.