On Simply Normal Numbers to Different Bases

TL;DR

This paper extends Schmidt's 1961/1962 result by characterizing and proving the existence of real numbers that are simply normal to specific sets of bases, with the set of such numbers having full Hausdorff dimension.

Contribution

It provides necessary and sufficient conditions for the existence of real numbers simply normal to certain bases and shows this set has full Hausdorff dimension, extending prior work on normal numbers.

Findings

Conditions on M are necessary and sufficient for such numbers to exist.

The set of these real numbers has full Hausdorff dimension.

Extends Schmidt's classical results to more general 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 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.