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arXiv · 1504.05121

Non-trivial matrix actions preserve normality for continued fractions

Joseph Vandehey
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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 $(ax+b)/(cx+d)$.

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Joseph Vandehey. 2015-04-20. Non-trivial matrix actions preserve normality for continued fractions. https://doi.org/10.1112/s0010437x16007740

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