A Mattila-Sj\"{o}lin theorem for triangles

We show for a compact set $E\subset {\mathbb{R}}^d$, $d\ge 4$, that if the Hausdorff dimension of $E$ is larger than $\frac{2}{3}d+1$, then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. Here we understand the set of congruence classes of triangles formed by triples of points of $E$ as the set $${\mathrm{\Delta }}_{\text{tri}}(E)=\{(t,r,\alpha ):\mathrm{\mid }x-z\mathrm{\mid }=t,\mathrm{\mid }y-z\mathrm{\mid }=r\text{ and }\alpha =\alpha (x,z,y),x,y,z\in E\},$$ where $\alpha (x,z,y)$ denotes the angle formed by $x$, $y$ and $z$ , centered at $z$. This extends the Mattila-Sj"{o}lin theorem that establishes a non-empty interior for the distance set instead of the set of congruence classes of triangles. These theorems can be thought of as refinements and extensions of the statements in the well known Falconer distance problem.