On the topological structure of the arithmetic sum of two Cantor sets

Abstract

This paper is concerned with the topological nature of the arithmetic sum of two Cantor sets. For the class of homogeneous Cantor sets, there are five possible structures for the sum: a Cantor set, a closed interval, or three mixed models called L, R and M-Cantorvals. Examples are given showing that all these possible structures actually occur. In the case of symmetric homogeneous Cantor sets, there are in fact only three possible topological types for the sum: a Cantor set, a closed interval or an M-Cantorval. For homogeneous Cantor sets generated by two intervals stronger results are given. Finally, an example is given showing that the result for homogeneous Cantor sets is not true in the more general class of the affine Cantor sets.