Topological structure of the sum of two affine Cantor sets

Abstract

We introduce a dense subset of affine Cantor sets, termed "generalized homogeneous Cantor sets." We show that for any two members of this class, if their sum set is not a Cantor set, it has a dense interior. If one of them is an affine Cantor set with two mappings, the same result holds. In sequel, we introduce a subclass of generalized homogeneous Cantor sets, denoted by $\mathcal{N}$, such that for any pair $K,K^{\prime }\in \mathcal{N}$, there are generically (i.e., from both topological and measure-theoretic) five possible structures for their sum set: a bilateral gap interval, an L, R, M-Cantorval, or a Cantor set. Finally, we explicitly present new pairs of affine Cantor sets which have stable intersection, while do not satisfy the Generalized Thickness Test.