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 b \ge 2 be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of b b -ary digits is a Sturmian sequence over { 0 , 1 , … , b − 1 } \{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 b -ary expansion of log ⁡ ( 1 + 1 a ) \log (1 + \frac {1}{a}) for any integer a ≥ 34 a \ge 34 .