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ON SIMULTANEOUSLY BADLY APPROXIMABLE NUMBERS

Published online by Cambridge University Press:  24 March 2003

ANDREW POLLINGTON
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USAandy@math.byu.edu
SANJU VELANI
Affiliation:
Department of Mathematics, Queen Mary, London, Mile End Road, London E1 4NS s.velani@qmw.ac.uk
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Abstract

For any pair $i,j\ge 0$ with $i+j=1$ let ${\mathbf Bad}(i,j)$ denote the set of pairs $(\alpha,\beta)\in {\bb R}^2$ for which $\max\{\|q\alpha\|^{1/i}\|q\beta\|^{1/j}\}>c/q$ for all $q\in {\bb N}$ . Here $c=c(\alpha,\beta)$ is a positive constant. If $i=0$ the set ${\mathbf Bad}(0, 1)$ is identified with ${\bb R}\times {\mathbf Bad}$ where ${\mathbf Bad}$ is the set of badly approximable numbers. That is, ${\mathbf Bad}(0, 1)$ consists of pairs $(\alpha, \beta)$ with $\alpha\in {\bb R}$ and $\beta\in {\mathbf Bad}$ If $j=0$ the roles of $\alpha$ and $\beta$ are reversed. It is proved that the set ${\mathbf Bad}(1,0)\cap {\mathbf Bad} (0,1)\cap {\mathbf Bad}(i,j)$ has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

Type
Notes and Papers
Copyright
The London Mathematical Society, 2002

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