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.
Categories: math.CA math.MG