Improvement on 2-Chains Inside Thin Subsets of Euclidean Spaces

Given E⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subset \mathbb {R}^d$$\end{document}, d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document}, we prove that when dimH(E)>d2+13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim _{{\mathcal {H}}}(E)>\frac{d}{2}+\frac{1}{3}$$\end{document}, the set of gaps of 2-chains inside E, i.e., Δ2(E)={(|x-y|,|y-z|):x,y,z∈E}⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset \mathbb {R}^2 \end{aligned}$$\end{document}has positive Lebesgue measure. This improves a result of Bennett, Iosevich, and Taylor. We also consider when the set of similarity classes of 2-chains, S2(E)=t1t2:(t1,t2)∈Δ2(E)=|x-y||y-z|:x,y,z∈E⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_2(E)=\left\{ \frac{t_1}{t_2}:(t_1,t_2)\in \Delta _2(E)\right\} =\left\{ \frac{|x-y|}{|y-z|}: x, y, z\in E \right\} \subset \mathbb {R} \end{aligned}$$\end{document}has positive Lebesgue measure. The main idea in this paper is to reduce geometric problems to integrals where Wolff-Erdoğan’s spherical averaging estimates apply. Invariant measures on orthogonal groups play an important role in the reduction.