On the expansions of a real number to several integer bases

Abstract

Very little is known about the expansions of a real number in several integer bases. We establish various results showing that the expansions of a real number in two multiplicatively independent bases cannot both be simple, in a suitable sense. We also construct explicitly a real number $\xi$ which is rich to all integer bases, that is, with the property that, for every integer $b\ge 2$, every finite block of letters in the alphabet $\{0,1,\dots ,b-1\}$ occurs in the $b$-ary expansion of $\xi$.