Non-trivial matrix actions preserve normality for continued fractions

Abstract

A seminal result due to Wall states that if $x$ is normal to a given base $b$ then so is $rx+s$ for any rational numbers $r,s$ with $r\ne 0$. We show that a stronger result is true for normality with respect to the continued fraction expansion. In particular, suppose $a,b,c,d\in \mathbb{Z}$ with $ad-bc\ne 0$. Then if $x$ is continued fraction normal, so is $(ax+b)/(cx+d)$.

References in corpus (4)

• Natural extensions and entropy of $\alpha$-continued fractions
• On the joint normality of certain digit expansions
• Distribution of approximants and geodesic flows
• Normality preserving operations for Cantor series expansions and associated fractals part II

Cited by in corpus (1)

• A Geometric Interpretation of the $p$-adic Littlewood Conjecture