Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma

TL;DR

This paper proves that certain finite point configurations in products of thick Cantor sets have non-empty interior, introducing a nonlinear version of the Newhouse gap lemma and demonstrating robustness of distance sets.

Contribution

It establishes a nonlinear gap lemma and shows that finite point configurations in product Cantor sets have interior, extending classical results to a nonlinear and robust setting.

Findings

Finite point configurations in product Cantor sets have non-empty interior.

Introduces a nonlinear version of the Newhouse gap lemma.

Distance sets are robust to small perturbations of the pin.

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 $\phi: \mathbb{R}^2\times \mathbb{R}^2 \rightarrow \mathbb{R} $ is a function satisfying some mild assumptions on its derivatives, then there exists an open set $S$ so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.