Intersections of Cantor sets with hyperbolas and continuous images
Yi Cai, Xiu Chen, Lipeng Wang
TL;DR
This work analyzes how products of middle Cantor sets $C_\lambda$ interact with hyperbolas and how their Cartesian products map under monomial-like transforms. It develops a fractal-interval framework using basic rank-$n$ intervals, a double covering set, and a binary-tree construction to obtain continuum many solutions to $xy=t$ when $0.4302\le\lambda<\tfrac{1}{2}$, proving $S_t$ has the cardinality of the continuum for all $t\in(0,1)$. It further identifies a family of thresholds $\lambda_k$ (roots of $(k-1)\lambda^k+2(k+1)\lambda-k-1=0$) ensuring $\{x^k y:x,y\in C_\lambda\}=[0,1]$ for $k\ge2$, with $\lambda_k$ increasing to $1/2$ as $k\to\infty$. The results reveal a delicate dependence on $\lambda$ and $k$ in the structure of fractal intersections and their continuous images, with extensions to other curves and open questions on optimal thresholds discussed in the final remarks.
Abstract
Given $λ\in (0,1/2)$, 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 $S_t=\set{(x,y)\in C_λ\times C_λ: xy=t}$, 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 $λ_0=0.4302$ such that for all $λ$ satisfying $λ_0 \le λ< 1/2$, the set $S_t$ has the cardinality of the continuum for every $t \in (0,1)$. Besides, we further investigate the continuous image of $C_λ\times C_λ$, that is, for any given $2\le k\in \nn$, we give a sufficient condition for set $\set{x^ky:x,y\in C_λ}$ 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 $λ$.