Non-trivial matrix actions preserve normality for continued fractions

TL;DR

This paper extends the understanding of normality preservation in continued fractions, showing that certain matrix transformations preserve continued fraction normality, unlike the linear case where only base normality is preserved.

Contribution

It proves that non-trivial matrix actions with integer entries preserve continued fraction normality, a stronger result than the classical linear case for base normality.

Findings

Matrix transformations with integer entries preserve continued fraction normality.

The result generalizes the linear case of normality preservation to a broader class of transformations.

Provides a new understanding of how continued fraction normality behaves under rational transformations.

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 $r\neq 0$. 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 $ad-bc\neq 0$. Then if $x$ is continued fraction normal, so is $(ax+b)/(cx+d)$.