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Non-trivial matrix actions preserve normality for continued fractions

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Elementary number theory

Published online by Cambridge University Press:  06 February 2017

Joseph Vandehey
Joseph Vandehey*
Affiliation:
Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210-1174, USA email [email protected]
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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 $(ax+b)/(cx+d)$.

Keywords

normal numbercontinued fractionergodic theory

MSC classification

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary: 11A55: Continued fractions
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 2 , February 2017 , pp. 274 - 293
Copyright
© The Author 2017 

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