On hinges inside thin subsets of Euclidean spaces

In this talk we will show that when the Hausdorff dimension of a subset $E \subset \mathbb{R}^d$ is greater than $\frac{d}{2}+\frac{1}{3}$, the Lebesgue measure of $$\{(|x-y|, |y-z|):x,y,z\in E\} $$ must be positive. It can be seen as a progress from the distance problem, where $\frac{d}{2}+\frac{1}{3}$ is known, to the pinned distance problem, where $\frac{d}{2}+\frac{1}{3}$ is expected.