Intersections of Siepinski gasket with its translation

Abstract

Let $E$ be the Sierpinski gasket, i.e., the self-similar set generated by the IFS $\left \{f_a(x)=\frac{x+a}{q}: a\in \{(0,0), (0,1), (1,0)\}\right \}$. In paper, we provide a description of the following set for $2<q<3$ \begin{equation*} D_q=\{\dim _H(E\cap (E+t)):\;t\in T\}, \end{equation*} where $T$ is the set of $t=(t_1, t_2)$ with $t\in E-E$ and $t_1, t_2$ have unique $q$-expansions w.r.t $\{-1,0,1\}$.