On the normality of numbers to different bases

Abstract

We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago. We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce.