Visibility of Cartesian products of Cantor sets

Let $K_λ$ be the attractor of the following IFS \begin{equation*} \{f_1(x)=λx, f_2(x)=λx+1-λ\}, \;\;0<λ<1/2. \end{equation*} Given $α\geq 0$, we say the line $y=αx$ is visible through $K_λ\times K_λ$ if $$ \{(x, αx): x\in \mathbb R\setminus \{0\}\}\cap ((K_λ\times K_λ))=\emptyset. $$ Let $V=\left \{α\geq 0: y=αx \mbox{ is visible through } K_λ\times K_λ \right \}$. 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.