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6

Normality to different bases

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 ...

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