Arithmetic on self-similar sets

Let K-1 and K-2 be two one-dimensional homogeneous self-similar sets with the same ratio of contractions. Let f be a continuous function defined on an open set U subset of R-2. Denote the continuous image of f by f(boolean OR)(K-1, K-2) = {f (x, y) : (x, y) is an element of (K-1 x K-2) boolean AND U}. In this paper we give a sufficient condition which guarantees that f(boolean OR)(K-1, K-2) contains some interiors. Our result is different from Simon and Taylor's (2020, Proposition 2.9) as we do not need the condition that the product of the thickness of K-1 and K-2 is strictly greater than 1. As a consequence, we give an application to the univoque sets in the setting of q-expansions. (C) 2020 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.