ON THE EXPANSIONS OF REAL NUMBERS IN TWO MULTIPLICATIVELY DEPENDENT BASES
Yann Bugeaud, Dong Han Kim · Bulletin of the Australian Mathematical Society · 2016
Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$ -ary expansion and the $s$ -ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$ , and we show that this bound is best possible.