THE DISCRIMINATION THEOREM FOR NORMALITY TO NONINTEGER 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 number s normal in base theta(r) are normal in base theta(s). Here theta is s ome 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 c omplete 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 th eta and are thus subject to the constraints imposed by the results of [7]. This paper then completes the answer to the problem raised by Men des France in [12] of determining those phi and psi for which normalit y in one implies normality in the other.