Average distances on self-similar sets and higher order average distances of self-similar measures

The purpose of this paper is twofold: (1) we study different notions of the average distance between two points of a self-similar subset of $${\mathbb {R}}$$ , and (2) we investigate the asymptotic behaviour of higher order average moments of self-similar measures on self-similar subsets of $${\mathbb {R}}$$ .