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 of tuples of edge lengths in $K_1\times K_2$ corresponding to a finite tree has non-empty interior, where $K_1,K_2\subset \mathbb{R}$ are Cantor sets of thickness $\tau (K_1)\cdot \tau (K_2)>1$. Our method relies on establishing that the pinned distance set is robust to small perturbations of the pin. In the process, we prove a nonlinear version of the classic Newhouse gap lemma, and show that if $K_1,K_2$ are as above and $\varphi :{\mathbb{R}}^2\times {\mathbb{R}}^2\to \mathbb{R}$ is a function satisfying some mild assumptions on its derivatives, then there exists an open set $S$ so that ${\bigcap }_{x\in S}\varphi (x,K_1\times K_2)$ has non-empty interior.