Revista Matemática Iberoamericana


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Volume 28, Issue 4, 2012, pp. 931–946
DOI: 10.4171/RMI/697

Published online: 2012-10-14

On the expansions of a real number to several integer bases

Yann Bugeaud[1]

(1) Université de Strasbourg, France

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 $\xi$.

Keywords: Digital expansion, normal number

Bugeaud Yann: On the expansions of a real number to several integer bases. Rev. Mat. Iberoamericana 28 (2012), 931-946. doi: 10.4171/RMI/697