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

TL;DR

This paper extends the Mattila-Sj"olin theorem to higher dimensions, showing that sets with sufficiently large Hausdorff dimension contain a rich set of congruence classes of simplices, including triangles in three dimensions.

Contribution

It provides new dimension thresholds ensuring the set of congruence classes of simplices has nonempty interior, extending previous results to all simplices and low dimensions.

Findings

Sets with high enough Hausdorff dimension have nonempty interior of simplex congruence classes.

Improves previous thresholds for the existence of rich simplex configurations in sets.

Extends results to include triangles in three dimensions and all simplices.

Abstract

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.