The boundaries of self-similar tiles in Rn

AbstractLet K be a self-similar set in Rn which has similarity dimension n and nonempty interior. In this paper it is shown that the topological boundary of K has Hausdorff dimension less than n. Examples are given to show that although the dimension of the boundary is strictly less than n, it may be arbitrarily close to n.Let K be a self-similar set in any complete metric space X such that K satisfies the strong open sets condition (SOSC). A recent result of A. Schief shows that dimHK=α where α is the similarity dimension of K. If O is the open set given by the SOSC, then it is shown in this paper that dimH(K\O)<α. More generally, if A is any inverse invariant closed subset of K, then dimHA