Topological structure of the sum of two homogeneous Cantor sets

Abstract

We show that in the context of homogeneous Cantor sets, there are generically five possible (open and dense) structures for their arithmetic sum: a Cantor set, an L, R, M-Cantorval and a finite union of closed intervals. The dense case has been dealt with previously. In this paper, we explicitly present pairs of this space which have stable intersection, while not satisfying the generalized thickness test. Also, all the pairs of middle homogeneous Cantor sets whose arithmetic sum is a closed interval are identified.

Characterization of the algebraic difference of special affine Cantor sets

“…Many authors were examining algebraic differences or sums of various types of Cantor sets (e.g. [1], [6], [8], [12], [13], [15], [16], [18], [19]). One can find its application for example in spectral theory (see [4], [20]) or number theory (see [7]).…”

“…However, studies on homogeneous and affine Cantor sets are still conducted by many researchers (e.g. [15], [21], [5]).…”