How likely can a point be in different Cantor sets

Let $m\in {\mathbb{N}}_{\ge 2}$, and let $\mathcal K={K_\lambda: \lambda\in(0, 1/m]}$beaclassofCantorsets,where$K_{\lambda}={\sum_{i=1}^\infty d_i\lambda^i: d_i\in{0,1,\ldots, m-1}, i\ge 1}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal{K}$. More precisely, for a fixed point $x\in (0,1)$ we consider the parameter set $\mathrm{\Lambda }(x)=\{\lambda \in (0,1\mathrm{/}m]:x\in K_{\lambda }\}$, and show that $\mathrm{\Lambda }(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $\mathrm{\Lambda }(x)$ we prove that the intersection $\mathrm{\Lambda }(x)\cap \mathrm{\Lambda }(y)$ also has full Hausdorff dimension for any $x, y\in(0,1)$.