Keeping in mind that almost all real numbers are normal to every integer base, we investigate the following question: Do there exist real numbers which are normal to one base r, but not normal to another base s? By Theorem 4.4 we know already that the answer is negative when r and s are multiplicatively dependent. However, at the end of the 1950s, Cassels and W. M. Schmidt, independently, gave a positive answer to this question when r and s are multiplicatively independent. Section 6.1 is devoted to their result. In the second section, we discuss its extension to non-integer bases. Then, we investigate what can be said on the expansions of a given number to two different bases. The final section is concerned with the study of the analogous question for representations of integers in two different bases.

Normality to a prescribed set of integer bases

Theorem 4.4, established by Maxfield [494], asserts that if r and s are multiplicatively dependent integers at least equal to 2, then a real number is normal to base r if, and only if, it is normal to base s. However, this result says nothing if r and s are multiplicatively independent. In ‘The new Scottish book’ (Problem 144), Steinhaus [663] asked whether normality with respect to infinitely many bases implies normality with respect to all other bases. Answers have been given independently by Cassels [182] and W. M. Schmidt [627]. Below is the statement established by Schmidt.