On the expansions of a real number to several integer bases
Yann Bugeaud · Revista Matemática Iberoamericana · 2012
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 .