The Structure of Arithmetic Sums of Affine Cantor Sets

Abstract

In this paper we describe the structure of the arithmetic sum of two affine Cantor sets. These are self-similar sets which are part of the dynamically defined Cantor sets. Let \({\textbf{C}_1}, {\textbf{C}_2}\) be affine Cantor sets with \([0,s]\) and \([0,r]\) as intervals of step 0. We show a generic family of these self-similar sets for which the structure of \({\textbf{C}_1}+{\textbf{C}_2}\) is of one of the following five types: (i) an \(M\)-Cantorval, (ii) an \(R\)-Cantorval, (iii) an \(L\)-Cantorval, or there exist \(\lambda, \eta >0\) and intervals \(I\), \({\tilde I}\) of the construction of \({\textbf{C}_1}\) and \({\textbf{C}_2}\), respectively, such that (iv) \(\lambda {\textbf{C}_1}+\eta {\textbf{C}_2}= {\textbf{C}_1}\cap I + {\textbf{C}_2}\cap {\tilde I} -\min I - \min {\tilde I} \,= \,[0, \lambda s + \eta r]\),\, or (v) \(\lambda {\textbf{C}_1}+\eta {\textbf{C}_2}= {\textbf{C}_1}\cap I + {\textbf{C}_2}\cap {\tilde I} -\min I - \min {\tilde I}\) is homeomorphic to the Cantor ternary set. This result generalizes the description obtained by Mendes and Oliveira for the case of homogeneous Cantor sets and the one obtained by the first two authors for semi-homogeneous Cantor sets. It also provides a suitable framework in which a question of Mendes and Oliveira admits a positive answer.