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

TL;DR

This paper proves that subsets of Euclidean space with positive upper Banach density contain large dilates of certain geometric configurations, extending known results to products of simplices and rectangles in higher dimensions.

Contribution

It extends the class of geometric configurations known to be contained in dense subsets of Euclidean space, including products of simplices and rectangles, with new proofs and dimension bounds.

Findings

Dense subsets contain large dilates of rectangles in dimensions ≥4.

Extension to products of two non-degenerate simplices in higher dimensions.

New proof of Bourgain's result on simplices in dense subsets.

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 $d\geq4$. 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 $d\geq k_1+k_2+6$ 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 $d\geq k+1$, a result originally due to Bourgain, is also presented.