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

Abstract:

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