A Mattila-Sj\"olin theorem for simplices in low dimensions

In this paper we show that if a compact set $E \subset \mathbb{R}^d$, $d \geq 3$, has Hausdorff dimension greater than $\frac{(4k-1)}{4k}d+\frac{1}{4}$ when $3 \leq d<\frac{k(k+3)}{(k-1)}$ or $d- \frac{1}{k-1}$ when $\frac{k(k+3)}{(k-1)} \leq d$, then the set of congruence class of simplices with vertices in $E$ has nonempty interior. By set of congruence class of simplices with vertices in $E$ we mean $$\Delta_{k}(E) = \left \{ \vec{t} = (t_{ij}) : |x_i-x_j|=t_{ij} ; \ x_i,x_j \in E ; \ 0\leq i < j \leq k \right \} \subset \mathbb{R}^{\frac{k(k+1)}{2}}$$ where $2 \leq k <d$. This result improves our previous work in the sense that we now can obtain a Hausdorff dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of $E$ has nonempty interior when $d=3$ as well as extending to all simplices. The present work can be thought of as an extension of the Mattila-Sj\"olin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.