On the expansions of real numbers in two integer bases

Abstract

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

11 pages, to appear at Annales de l'Institut Fourier