On the sum of two affine Cantor sets

Suppose that $K$ and $K^{\mathrm{\prime }}$ are two affine Cantor sets. It is shown that the sum set $K+K^{\mathrm{\prime }}$ has equal box and Hausdorff dimensions and in this number named $s$, $H^s(K+K^{\mathrm{\prime }})<\mathrm{\infty }$. Moreover, for almost every pair $(K,K^{\mathrm{\prime }})$ satisfying $HD(K)+HD(K^{\mathrm{\prime }})\le 1$, there is a dense subset $D\subset \mathbb{R}$ such that $H^s(K+\lambda K^{\mathrm{\prime }})=0$, for all $\lambda \in D$. It also is shown that in the context of affine Cantor sets with two increasing maps, there are generically (topological and almost everywhere) five possible structures for their sum: a Cantor set, an L, R, M-Cantorval or a finite union of closed intervals.