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Separation properties for self-similar sets. (English) Zbl 0807.28005
Summary: Given a self-similar set \(K\) in \(\mathbb{R}^ s\) we prove that the strong open set condition and the open set condition are both equivalent to \(H^ \alpha(K)> 0\), where \(\alpha\) is the similarity dimension of \(K\) and \(H^ \alpha\) denotes the Hausdorff measure of this dimension. As an application we show for the case \(\alpha= s\) that \(K\) possesses inner points iff it is not a Lebesgue null set.

28A78 Hausdorff and packing measures
28A80 Fractals
Full Text: DOI
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