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 $${\Delta }_k(E)=\left\{\vec{t}=(t_{ij}):|x_i-x_j|=t_{ij};x_i,x_j\in E;0\le i<j\le k\right\}\subset {\mathbb{R}}^{\frac{k(k+1)}{2}}$$ where $2\le k<d$. 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-Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.

arXiv comment20 pages, 3 figures