A Mattila-Sj\"{o}lin theorem for triangles

TL;DR

This paper extends the Mattila-Sj"{o}lin theorem to show that for sets in high-dimensional space with sufficiently large Hausdorff dimension, the set of all congruence classes of triangles formed by points in the set has a nonempty interior, refining the Falconer distance problem.

Contribution

It generalizes the Mattila-Sj"{o}lin theorem from distances to triangle congruence classes in higher dimensions for sets with large Hausdorff dimension.

Findings

Set of triangle congruence classes has nonempty interior for large Hausdorff dimension

Extends Mattila-Sj"{o}lin theorem from distances to triangles

Provides new insights into geometric measure theory and Falconer distance problem

Abstract

We show for a compact set $E \subset \mathbb{R}^d$, $d \geq 4$, that if the Hausdorff dimension of $E$ is larger than $\frac{2}{3}d+1$, then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. Here we understand the set of congruence classes of triangles formed by triples of points of $E$ as the set $$\Delta_{\text{tri}}(E) = \left \{ (t,r, \alpha) : |x-z|=t, |y-z|=r \, \text{ and }\, \alpha= \alpha(x,z,y), \ x,y,z \in E \right \},$$ where $\alpha (x,z,y)$ denotes the angle formed by $x$, $y$ and $z$ , centered at $z$. This extends the Mattila-Sj\"{o}lin theorem that establishes a non-empty interior for the distance set instead of the set of congruence classes of triangles. These theorems can be thought of as refinements and extensions of the statements in the well known Falconer distance problem.