The discrimination theorem for normality to non-integer bases

In a previous paper, [7], the authors together with Gavin Brown gave a complete description of the values ofθ, r ands for which numbers normal in baseθr are normal in baseθs. Hereθ is some real number greater than 1 andx is normal in baseθ if {θnx} is uniformly distributed modulo 1. The aim of this paper is to complete this circle of ideas by describing thoseφ andψ for which normality in baseφ implies normality in baseψ. We show, in fact, that this can only happen if both are integer powers of some baseθ and are thus subject to the constraints imposed by the results of [7]. This paper then completes the answer to the problem raised by Mendès France in [12] of determining thoseφ andψ for which normality in one implies normality in the other.