Improvement on $2$-chains inside thin subsets of Euclidean spaces

TL;DR

This paper demonstrates that sets in Euclidean spaces with Hausdorff dimension exceeding certain thresholds contain rich geometric configurations of 2-chains, with positive measure of their gap sets and similarity classes, extending previous results in geometric measure theory.

Contribution

It improves the known dimension thresholds for the positivity of gap sets and similarity classes of 2-chains within thin subsets of Euclidean spaces, generalizing and strengthening prior results.

Findings

Positive Lebesgue measure of 2-chain gap sets for dim > d/2 + 1/3

Positive measure of similarity classes for dim > d/2 + 1/7

Generalization of Wolff-Erdogan's distance result and improvement of Bennett-Iosevich-Taylor's chain result

Abstract

We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\Delta_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has positive Lebesgue measure. It generalizes Wolff-Erdogan's result on distances and improves a result of Bennett, Iosevich and Taylor on finite chains. We also consider the similarity class of $2$-chains, $$S_2(E)=\left\{\frac{t_1}{t_2}:(t_1,t_2)\in\Delta_2(E)\right\}=\left\{\frac{|x-y|}{|y-z|}: x, y, z\in E \right\}\subset\mathbb{R},$$ and show that $|S_2(E)|>0$ whenever $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{7}$.