Fractal Structure of Parametric Cantor Sets With a Common Point

For $\lambda >0$, let $E_{\lambda }$ be the self-similar set generated by the iterated function system (IFS) $\{\frac{x}{3},\frac{x+\lambda }{3}\}$. In this paper we study the structure of parameters $\lambda$ in which $E_{\lambda }$ contains a common point. $E_{\lambda }$. More precisely, for a given point $x>0$ we consider the topology of the parameter set $\mathrm{\Lambda }\left(x\right)=\{\lambda >0:x\in E_{\lambda }\}$. We show that $\mathrm{\Lambda }\left(x\right)$ is a Lebesgue null set contains neither interior points nor isolated points, and the Hausdorff dimension of $\mathrm{\Lambda }\left(x\right)$ is $\log 2\mathrm{/}\log 3$. Furthermore, we consider the set ${\mathrm{\Lambda }}_{\mathrm{n}\mathrm{o}\mathrm{t}}(x)$ which consists of all parameters $\lambda$ that the digit frequency of $x$ in base $\lambda$ does not exist. We also consider the set ${\mathrm{\Lambda }}_p(x)$ consisting of all $\lambda$ in which the digit frequency of $2$ in the base $\lambda$ expansion of $x$ is $p$. We show that the Hausdorff dimension of ${\mathrm{\Lambda }}_{\mathrm{n}\mathrm{o}\mathrm{t}}\left(x\right)$ is $\log 2\mathrm{/}\log 3$ and the lower bound Hausdorff dimension of ${\mathrm{\Lambda }}_p\left(x\right)$ is $-p{\log }_{3}p-(1-p){\log }_3(1-p)$.