On the joint normality of certain digit expansions

Abstract

We prove that a point $x$ is normal with respect to an ergodic, number-theoretic transformation $T$ if and only if $x$ is normal with respect to $T^n$ for any $n\ge 1$. This corrects an erroneous proof of Schweiger. Then, using some insights from Schweiger's original proof, we extend these results, showing for example that a number is normal with respect to the regular continued fraction expansion if and only if it is normal with respect to the odd continued fraction expansion.