Hausdorff dimension of the arithmetic sum of self-similar sets

Let beta > 1. We define a class of similitudes S := { f(i) (x) = x/beta(n)(i) + a(i) : n(i) is an element of N+, a(i) is an element of R}. Taking any finite collection of similitudes {f(i) (x)}(i=1)(m) from S, it is well known that there is a unique self similar set K-1 satisfying K-1 = U(i=1)(m)f(i) (K-1). Similarly, another self-similar set K-2 can be generated via the finite contractive maps of S. We call K-1 + K-2 = {x + y : x is an element of K-1, y is an element of K-2) the arithmetic sum of two self-similar sets. In this paper, we prove that K-1 + K-2 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 K-1 + K-2 under some conditions, which partially provides the dimensional result of K-1 + K-2 if the IFS's of K-1 and K-2 fail the irrationality assumption, see Peres and Shmerkin (2009). (C) 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.