How inhomogeneous Cantor sets can pass a point

Abstract

For $x >0$, let

$$\begin{aligned} \Upsilon (x) = \left\{ (a,b): x\in E_{a,b}, a>0, b>0, a+b \le 1 \right\} , \end{aligned}$$

where $E_{a,b}$ is the unique nonempty compact invariant set generated by the inhomogeneous IFS

$$\begin{aligned} \Psi _{a,b} = \left\{ f_0(x)= a x,\; f_1(x)= b(x+1) \right\} . \end{aligned}$$

We show that the set $\Upsilon (x)$ is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets $\Upsilon (x_1),\ldots , \Upsilon (x_\ell )$ still has full Hausdorff dimension for any finite number of positive numbers $x_1, \ldots , x_\ell$.