A geometric property of certain plane sets
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- by Kenneth Pietz PDF
- Proc. Amer. Math. Soc. 54 (1976), 197-200 Request permission
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.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 197-200
- DOI: https://doi.org/10.1090/S0002-9939-1976-0393429-8
- MathSciNet review: 0393429