`The irrationality exponent of real numbers and the expansion in integer bases'

Abstract:
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. Let r and s be multiplicatively independent positive
integers. We establish that the r-ary expansion and the s-ary
expansion of an irrational real number, viewed as infinite words on
{0,1,...,r − 1} and {0,1,...,s − 1}, respectively, cannot have
simultaneously a low block complexity. In particular, they cannot be
both Sturmian words.
This talk is based on joint work with Yann Bugeaud.