We introduce and study a new complexity function in combinatorics on
words, which takes into account the smallest return 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.
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. If the
irrationality exponent of $\xi$ is equal to 2 or slightly greater than
2, then the b-ary expansion of $\xi$ cannot be `too simple', in a
suitable sense. Our result applies, among other classical numbers, to
badly approximable numbers, non-zero rational powers of $e$, and
$\log(1+1/a)$, provided that the integer a is sufficiently large. It
establishes an unexpected connection between the irrationality
exponent of a real number and its b-ary expansion.
This is joint work with Yann Bugeaud.