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arXiv · 2111.09393

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

Alex McDonald·Krystal Taylor
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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 $τ(K_1)\cdot τ(K_2) >1$. 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 $ϕ: \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 $\bigcap_{x \in S} ϕ(x,K_1\times K_2)$ has non-empty interior.

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Alex McDonald, Krystal Taylor. 2021-11-17. Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma. https://arxiv.org/abs/2111.09393

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