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

We prove that if the Hausdorff dimension of $E\subset {\mathbb{R}}^d$, $d\ge 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $${\mathrm{\Delta }}_2(E)=\{(\mathrm{\mid }x-y\mathrm{\mid },\mathrm{\mid }y-z\mathrm{\mid }):x,y,z\in E\}\subset {\mathbb{R}}^2$$ 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, $$S_2(E)=\{\frac{t_1}{t_2}:(t_1,t_2)\in {\mathrm{\Delta }}_2(E)\}=\{\frac{\mathrm{\mid }x-y\mathrm{\mid }}{\mathrm{\mid }y-z\mathrm{\mid }}:x,y,z\in E\}\subset \mathbb{R},$$ and show that $\mathrm{\mid }{S}_2(E)\mathrm{\mid }>0$ whenever ${\dim }_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{7}$.