The boundaries of self-similar tiles in Rn

Let 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. 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<α.