On the expansions of real numbers in two multiplicative dependent bases

Abstract

Let $r\ge 2$ and $s\ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\dots ,r-1\}$ and $\{0,1,\dots ,s-1\}$, and we show that this bound is best possible.

arXiv comment15pages. arXiv admin note: substantial text overlap with arXiv:1512.06935