A Mattila–Sjölin theorem for simplices in low dimensions

Abstract

In this paper we show that if a compact set \(E \subset {\mathbb {R}}^d\), \(d \ge 3\), has Hausdorff dimension greater than \(\frac{(4k-1)}{4k}d+\frac{1}{4}\) when \(3 \le d<\frac{k(k+3)}{(k-1)}\) or \(d- \frac{1}{k-1}\) when \(\frac{k(k+3)}{(k-1)} \le d\), 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

$$\begin{aligned} \Delta _{k}(E) = \left\{ \textbf{t} = \left( t_{ij} \right) : |x_i-x_j|=t_{ij}; \ x_i,x_j \in E; \ 0\le i < j \le k \right\} \subset {\mathbb {R}}^{\frac{k(k+1)}{2}} \end{aligned}$$

where \(2 \le k <d\). This result improves the previous best results 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–Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.