How likely can a point be in different Cantor sets

Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_λ: λ\in(0, 1/m]\}$ be a class of Cantor sets, where $K_λ=\{\sum_{i=1}^\infty d_iλ^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. 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 $Λ(x)=\{λ\in(0,1/m]: x\in K_λ\}$, and show that $Λ(x)$ 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 $Λ(x)$ we prove that the intersection $Λ(x)\capΛ(y)$ also has full Hausdorff dimension for any $x, y\in(0,1)$.

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