On the arithmetic sums of Cantor sets

Abstract

Let $C_\la$ and $C_\ga$ be two affine Cantor sets in $\mathbb{R}$ with similarity dimensions $d_\la$ and $d_\ga$, respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having $\Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0$ where $C_\la + C_\ga$ denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form $C_\la \times C_\ga$. We prove that for Lebesgue almost all directions $\theta$ for which the projection is not one-to-one, the projection has zero $(d_\la + d_\ga)$-dimensional Hausdorff measure. We demonstrate the results on the case when $C_\la$ and $C_\ga$ are the middle-$(1-2\la)$ and middle-$(1-2\ga)$ sets.