On the Computational Problems of Upper Convex Densities for Self-Similar Sets with the Open Set Condition

Abstract

Let E be a self-similar set satisfying the open set condition. Zhou and Feng posed an open problem in 2004 as follows: x∈E, under what conditions is there a set U_x containing x with $\left|U_{x}\right|>0$ such that $\bar{D}_{C}^{s}(E, x)=\frac{\mathcal{H}^{s}\left(E \cap U_{x}\right)}{\left|U_{x}\right|^{s}}$? The aim of this paper is to present a solution of this problem. Under the assumption that there exists a nonempty convex open set containing E and satisfying the requirement of the open set condition, it is proved that if x ∈ E and the upper convex density of E at x equals 1, then there exists a convex set U_x containing x with $\left|U_{x}\right|>0$ such that $\bar{D}_{C}^{s}(E, x)= \frac{\mathcal{H}^{s}\left(E \cap U_{x}\right)}{\left|U_{x}\right|^{s}}$. Finally, as an application of this result, an equivalent condition for $E_{0}=E$ is given, where $E_{0}=\left\{x \in E \mid \bar{D}_{C}^{s}(E, x)=1\right\}$.

Keywords

Hausdorff measure / self-similar set / open set condition / upper convex density