Intersections of Cantor sets with hyperbolas and continuous images

Given $\lambda\in (0,1/2)$, let \begin{equation*} C_\lambda=\set{(1-\lambda)\sum_{i=1}^\infty d_i\lambda^{i-1}:d_i\in\set{0,1}} \end{equation*} be the middle Cantor sets with convex hull $[0, 1]$. We are interested in the set $S_t=\set{(x,y)\in C_\lambda\times C_\lambda: xy=t}$, where $t\in[0,1]$. Since the cases where $t=0$ or $t=1$ are trivial, we assume that $t\in(0,1)$ in what follows. We show that there exists a $\lambda_0=0.4302$ such that for all $\lambda$ satisfying $\lambda_0 \le \lambda < 1/2$, the set $S_t$ has the cardinality of the continuum for every $t \in (0,1)$. Besides, we further investigate the continuous image of $C_\lambda\times C_\lambda$, that is, for any given $2\le k\in \nn$, we give a sufficient condition for set $\set{x^ky:x,y\in C_\lambda}$ to be the interval $[0,1]$. Our observations reveal that the behavior exhibited by the image of the function $f_k(x,y)=x^ky$ is complex and depends on the parameters $k$ and $\lambda$.