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

TL;DR

This paper extends Bourgain's result by demonstrating that in any measurable subset of  with positive upper density, all proper k-degenerate distance graphs with d  can be embedded, generalizing previous configurations.

Contribution

It provides a sharp extension of Bourgain's theorem to all proper k-degenerate distance graphs in  for sets with positive upper density.

Findings

All proper k-degenerate distance graphs can be embedded in sets of positive upper density in .

The result applies for all d  k+1, extending previous configurations.

The extension is sharp and covers a broader class of geometric configurations.

Abstract

We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance graphs.