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 of theta, r and s for which numbers normal in base theta(r) are normal in base theta(s). Here theta is some real number greater than 1 and x is normal in base theta if {theta(n)x} is uniformly distributed module 1. The aim of this paper is to complete this circle of ideas by describing those phi and psi for which normality in base phi implies normality in base psi. We show, in fact, that this can only happen if both are integer powers of some base theta and are thus subject to the constraints imposed by the results of [7]. This paper then completes the answer to the problem raised by Mendes France in [12] of determining those phi and psi for which normality in one implies normality in the other.