On the expansions of real numbers in two integer bases

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 $\{0,1,\dots ,r-1\}$ and ${0, 1, \ldots , s-1}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.