Visibility of Cartesian products of Cantor sets

Abstract

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

13 pages