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

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 $d\ge k+2$ and $k$ is any positive integer.

Minor revision made. To appear in Proc. Amer. Math. Soc