Projections of cartesian products of the self-similar sets without the irrationality assumption

Let $\beta>1$. Define a class of similitudes \[S=\left\{f_{i}(x)=\dfrac{x}{\beta^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Let $\mathcal{A}$ be the collection of all the self-similar sets generated by the similitudes from $S$. In this paper, we prove that for any $\theta\in(0,\pi)$ and $K_1, K_2\in \mathcal{A}$, $Proj_{\theta}(K_1\times K_2)$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $Proj_{\theta}$ denotes the orthogonal projection onto $L_{\theta}$, and $L_{\theta}$ denotes the line through the origin in direction $\theta$. As a corollary, $\dim_{P}(Proj_{\theta}(K_1\times K_2))=\overline{\dim}_{B}(Proj_{\theta}(K_1\times K_2))$ holds for any $\theta\in(0,\pi)$ and any $K_1, K_2\in \mathcal{A}$, where $\dim_{P}$ and $\overline{\dim}_{B}$ denote the packing and upper box dimension. Whether $Proj_{\theta}(K_1\times K_2)$ is similar to a self-similar set or not is uniquely determined by the similarity ratios of $K_1$ and $K_2$ rather than the angle $\theta.$ When $Proj_{\theta}(K_1\times K_2)$ is similar to a self-similar set, in terms of the finite type condition \cite{NW}, we are able to calculate in cerntain cases the Hausdorff dimension of $Proj_{\theta}(K_1\times K_2)$. If $Proj_{\theta}(K_1\times K_2)$ is similar to an attractor of some infinite iterated function system, then by virtue of the Vitali covering lemma \cite{FG} we give an estimation of the Hausdorff dimension of $Proj_{\theta}(K_1\times K_2)$. For some cases, we can calculate, by means of Mauldin and Urbanski' result \cite{MRD}, the exact Hausdorff dimension of $Proj_{\theta}(K_1\times K_2)$. We also find some non-trivial examples such that for some angle $\theta\in[0,\pi)$ and some $K_1, K_2\in \mathcal{A}$, $\dim_{H}(Proj_{\theta}(K_1\times K_2))=\dim_{H}(K_1)+\dim_{H}(K_2)$.