Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma

Abstract 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 more general results where distance map $|x-y|$ replaced by function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. process, establish nonlinear version classic Newhouse Gap Lemma, and show if $K_1,K_2, \phi$ are as above then there exists an open S so $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ interior.