Hausdorff dimension of the arithmetic sum of self-similar sets

Let $\beta >1$. We define a class of similitudes $$S:=\{f_i(x)={\displaystyle \frac{x}{{\beta }^{n_i}}}+a_i:n_i\in {\mathbb{N}}^{+},a_i\in \mathbb{R}\}\mathrm{.}$$ Taking any finite similitudes $\{f_i(x){\}}_{i=1}^m$ from $S$, it is well known that there is a unique self-similar set $K_1$ satisfying $K_1={\cup }_{i=1}^m{f}_i(K_1)$. Similarly, another self-similar set $K_2$ can be generated via the finite contractive maps of $S$. We call $K_1+K_2=\{x+y:x\in K_1,y\in K_2\}$ the arithmetic sum of two self-similar sets. In this paper, we prove that $K_1+K_2$ is either a self-similar set or a unique attractor of some infinite iterated function system. Using this result we can then calculate the exact Hausdorff dimension of $K_1+K_2$ under some conditions, which partially provides the dimensional result of $K_1+K_2$ if the IFS’s of $K_1$ and $K_2$ fail the irrational assumption, see Peres and Shmerkin \cite{PS}.