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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 aninfinite word. We characterize the eventually periodic words and the Sturmianwords by means of this function. Then, we establish a new result on repetitionsin Sturmian words and show that it is best possible. Let $b \ge 2$ be aninteger. We deduce a lower bound for the irrationality exponent of real numberswhose 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 $a \ge 34$.

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