Hausdorff dimension of the arithmetic sum of self-similar sets

Let β>1β>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 SS, it is well known that there is a unique self-similar set K1K1 satisfying K1=∪i=1mfi(K1). Similarly, another self-similar set K2K2 can be generated via the finite contractive maps of SS. We call K1+K2={x+y:x∈K1,y∈K2}K1+K2={x+y:x∈K1,y∈K2} the arithmetic sum of two self-similar sets. In this paper, we prove that K1+K2K1+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+K2K1+K2 under some conditions, which partially provides the dimensional result of K1+K2K1+K2 if the IFS’s of K1K1 and K2K2 fail the irrationality assumption, see Peres and Shmerkin (2009).