On a class of self-similar sets which contain finitely many common points

Abstract: For $\lambda\in(0,1/2]$ let $K_\lambda \subset\mathbb{R}$ be a self-similar set generated by the iterated function system ${\lambda x, \lambda x+1-\lambda}$. Given $x\in(0,1/2)$, let $\Lambda(x)$ be the set of $\lambda\in(0,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $\lambda\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_\lambda$.