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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Separation properties for self-similar sets


Author: Andreas Schief
Journal: Proc. Amer. Math. Soc. 122 (1994), 111-115
MSC: Primary 28A80; Secondary 28A78
MathSciNet review: 1191872
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Abstract: 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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1191872-1
PII: S 0002-9939(1994)1191872-1
Keywords: Self-similar sets, fractals, Hausdorff dimension
Article copyright: © Copyright 1994 American Mathematical Society