A geometric property of certain plane sets

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.