Visibility of Cartesian products of Cantor sets

Abstract

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 $\alpha \geq 0$, we say the line $y=\alpha x$ is visible through $K_{\lambda}\times K_{\lambda}$ if $$ \{(x, \alpha x): x\in \mathbb R\setminus \{0\}\}\cap ((K_{\lambda}\times K_{\lambda}))=\emptyset. $$ 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.