The boundaries of self-similar tiles in \(\mathbb{R}^n\) (Q1296289)

This paper is devoted to the Hausdorff dimension of the topological boundary of a self-similar set \(K\subseteq \mathbb{R}^n\). The author shows that the Hausdorff dimension of \(K\) must be less than \(n\). Moreover, he gives examples showing that the Hausdorff dimension of the topological boundary of \(K\) can be arbitrarily close to \(n\). He also considers more general self-similar sets in a complete metric space.