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ON THE DIMENSIONLESSNESS OF INVARIANT SETS

Published online by Cambridge University Press:  10 September 2003

L. OLSEN
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland e-mail: lo@st-and.ac.uk
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Abstract

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Let $M$ be a subset of $\Bbb R$ with the following two invariance properties: (1) $M+k\subseteq M$ for all integers $k$, and (2) there exists a positive integer $l\ge 2$ such that $\frac{1}{l}M\subseteq M$. (For example, the set of Liouville numbers and the Besicovitch-Eggleston set of non-normal numbers satisfy these conditions.) We prove that if $h$ is a dimension function that is strongly concave at $0$, then the $h$-dimensional Hausdorff measure $\cal H^{h}(M)$ of $M$ equals $0$ or infinity.

Keywords

Type
Research Article
Copyright
© 2003 Glasgow Mathematical Journal Trust