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

For $λ\in(0,1/2]$ let $K_λ\subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{λx, λx+1-λ\}$. Given $x\in(0,1/2)$, let $Λ(x)$ be the set of $λ\in(0,1/2]$ such that $x\in K_λ$. In this paper we show that $Λ(x)$ 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 $λ\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_λ$.