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