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

Abstract

We establish that any subset of ${\mathbb{R}}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided $d\ge 4$. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if ${\Delta }_{k_1}$ and ${\Delta }_{k_2}$ are two fixed non-degenerate simplices of $k_1+1$ and $k_2+1$ points respectively, then any subset of ${\mathbb{R}}^d$ of positive upper Banach density with $d\ge k_1+k_2+6$ will necessarily contain an isometric copy of all sufficiently large dilates of ${\Delta }_{k_1}\times {\Delta }_{k_2}$. A new direct proof of the fact that any subset of ${\mathbb{R}}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of $k+1$ points provided $d\ge k+1$, a result originally due to Bourgain, is also presented.

Substantial revisions made. To appear in Math. Proc. Cambridge Philos. Soc

Cited by in corpus (1)

• On side lengths of corners in positive density subsets of the Euclidean space