Simplices and sets of positive upper density in $\mathbb{R}^d$

TL;DR

This paper extends Bourgain's theorem on distances to higher-dimensional simplices, showing that sets with positive upper density in Euclidean space contain many such simplices under certain dimensional conditions.

Contribution

It generalizes Bourgain's theorem from distances to arbitrary non-degenerate simplices in higher dimensions for sets with positive upper density.

Findings

Extension of Bourgain's theorem to simplices in 

Sets of positive upper density contain non-degenerate simplices

Applicable for dimensions d  xceeding k+2

Abstract

We prove an extension of Bourgain's theorem on pinned distances in measurable subset of $\mathbb{R}^2$ of positive upper density, namely Theorem $1^\prime$ in [Bourgain, 1986], to pinned non-degenerate $k$-dimensional simplices in measurable subset of $\mathbb{R}^{d}$ of positive upper density whenever $d\geq k+2$ and $k$ is any positive integer.