In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$. We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.

在本文中，我们证明集合$\{|x^1-x^2|,\dots,|x^kx^{k+1}|\,{:}\,x^i\in E\}当$ E\subset \mathbb{R}^2$是厚康托集$K_1,K_2\subset\mathbb{R}$的笛卡尔积时， $在 $\mathbb{R}^k$ 中具有非空内部。我们还证明了更一般的结果，其中距离图$|xy|$被函数$\phi(x,y)$取代，满足其偏导数的温和假设。在此过程中，我们建立了经典 Newhouse Gap Lemma 的非线性版本，并证明如果$K_1,K_2, \phi$如上，则存在一个开集S，使得$\bigcap_{x \in S} \ phi(x,K_1\乘以K_2)$具有非空的内部。