  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 X be the set of positive integers that are not
perfect powers, hence X is the set {2,3, 5,6,7,10,11,...} . Let M be a function
from X to sets of positive integers such that, for each s in X, 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 s^m such that s
is in X 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.
