Visibility of Cartesian products of Cantor sets

Let $K_{\lambda }$ be the attractor of the following IFS \begin{equation*} {f_1(x)=\lambda x, f_2(x)=\lambda x+1-\lambda}, ;;0<\lambda<1/2. \end{equation*} Given $\alpha \ge 0$, we say the line $y=\alpha x$ is visible through $K_{\lambda }\times K_{\lambda }$ if $$\{(x,\alpha x):x\in \mathbb{R}\setminus \{0\}\}\cap ((K_{\lambda }\times K_{\lambda }))=\mathrm{\varnothing }\mathrm{.}$$ Let $V=\left {\alpha \geq 0: y=\alpha x \mbox{ is visible through } K_{\lambda}\times K_{\lambda} \right }$. In this paper, we give a completed description of $V$, e.g., its Hausdoff dimension and its topological property. Moreover, we also discuss another type of visible problem which is related to the slicing problem.