Improvement on $2$-chains inside thin subsets of Euclidean spaces (1709.06814v2)

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 $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{7}$.