Geodesic
ON THE DIMENSIONLESSNESS OF INVARIANT SETS
Glasgow mathematical journal, Tome 45 (2003) no. 3, pp. 539-543

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DOI

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.
DOI : 10.1017/S0017089503001447
Mots-clés : 28A80 
OLSEN, L. ON THE DIMENSIONLESSNESS OF INVARIANT SETS. Glasgow mathematical journal, Tome 45 (2003) no. 3, pp. 539-543. doi: 10.1017/S0017089503001447
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     journal = {Glasgow mathematical journal},
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     year = {2003},
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     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089503001447/}
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