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

A geometric property of certain plane sets


Author: Kenneth Pietz
Journal: Proc. Amer. Math. Soc. 54 (1976), 197-200
MathSciNet review: 0393429
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Abstract: Suppose $ K$ is a compact subset of the plane of the form $ \overline {\Delta (0,1)} \backslash \cup _{n = 1}^\infty \Delta ({p_n},{r_n})$ where $ \overline {\Delta ({p_n},{r_n})} \subseteq \Delta (0,1)$ for each $ n$ and $ \overline {\Delta ({p_i},{r_i})} \cap \overline {\Delta ({p_j},{r_j})} = \emptyset $ for $ i \ne j$. Let $ \alpha = {\sup _{i \geqslant 1}}(({r_i} + 1)/{r_i})$ and define the sets $ {\partial _{\ast}}K \equiv \partial \Delta (0,1) \cup [ \cup _{n = 1}^\infty \partial \Delta ({p_n},{r_n})]$ and $ F(K) \equiv \{ z \in K\backslash {\partial _{\ast}}K = z{\text{ is not a point of density of }}K\} $. It is proved that if $ \alpha < 1$, then $ {\mathcal{K}^1}[F(K)] = 0$, where $ {\mathcal{K}^1}$ denotes Hausdorff one-dimensional measure.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0393429-8
PII: S 0002-9939(1976)0393429-8
Article copyright: © Copyright 1976 American Mathematical Society