Hausdorff dimension of the arithmetic sum of self-similar sets

Abstract

Let β>1. We define a class of similitudes S:=(fi(x)=xβni+ai:ni∈N+,ai∈R). Taking any finite collection of similitudes (fi(x))i=1m from S, it is well known that there is a unique self-similar set K1 satisfying K1=∪i=1mfi(K1). Similarly, another self-similar set K2 can be generated via the finite contractive maps of S. We call K1+K2=(x+y:x∈K1,y∈K2) the arithmetic sum of two self-similar sets. In this paper, we prove that K1+K2 is either a self-similar set or a unique attractor of some infinite iterated function system. Using this result we can calculate the exact Hausdorff dimension of K1+K2 under some conditions, which partially provides the dimensional result of K1+K2 if the IFS's of K1 and K2 fail the irrationality assumption, see Peres and Shmerkin (2009).