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 $0, 1, \ldots , s-1$ occurs with the same frequency $1/s$. Let ${\mathcal{S}}$ be the set of positive integers that are not perfect powers, hence $\mathcal{S}$ is the set $\{2,3, 5,6,7,10,11,\ldots \}$. Let $M$ be a function from $\mathcal{S}$ to sets of positive integers such that, for each $s$ in $\mathcal{S}$, 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 $s^m$ such that $s$ is in $\mathcal{S}$ 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.