Intersections of Cantor sets with hyperbolas and continuous images

Abstract

Given $\hat{I}»\in (0,1/2)$, let \begin{equation*} C_λ=\set{(1-λ)\sum_{i=1}^\infty d_iλ^{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 $S_t=\{(x,y)\in C_{\hat{I}}»\times C_{\hat{I}}»:xy=t\}$, 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 $\hat{I}{»}_0=0.4302$ such that for all $\hat{I}»$ satisfying $\hat{I}{»}_0\le \hat{I}»<1/2$, the set $S_t$ has the cardinality of the continuum for every $t\in (0,1)$. Besides, we further investigate the continuous image of $C_{\hat{I}}»\times C_{\hat{I}}»$, that is, for any given $2\le k\in \nn$, we give a sufficient condition for set $\{x^{k}y:x,y\in C_{\hat{I}}»\}$ to be the interval $[0,1]$. Our observations reveal that the behavior exhibited by the image of the function $f_k(x,y)=x^{k}y$ is complex and depends on the parameters $k$ and $\hat{I}»$.