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 which is rich to all integer bases, that is, with the property that, for every integer , every finite block of letters in the alphabet occurs in the -ary expansion of .
Cite this article
Yann Bugeaud, On the expansions of a real number to several integer bases. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 931–946DOI 10.4171/RMI/697