On simply normal numbers to different bases

Abstract

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0,1,…,s-1 occurs with the same frequency 1/s. Let S be the set of positive integers that are not perfect powers, hence S is the set {2,3,5,6,7,10,11,…}. Let M be a function from S to sets of positive integers such that, for each s in S, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases sm such that s is in S and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.