Boxes, extended boxes, and sets of positive upper density in the Euclidean space

TL;DR

This paper proves that large sets with positive upper Banach density in high-dimensional Euclidean spaces contain scaled copies of certain geometric patterns, extending previous Euclidean density theorems.

Contribution

It introduces new results showing that dense sets contain large dilates of boxes, extended boxes, and three-point corners, generalizing earlier theorems in Euclidean combinatorics.

Findings

Sets with positive upper Banach density contain large dilates of specific patterns.

The results unify and extend several Euclidean density theorems.

Patterns include vertices of boxes, extended boxes, and three-point corners.

Abstract

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed $n$-dimensional rectangular box, the same vertices extended with $n$ points completing three-term arithmetic progressions, and the same vertices extended with $n$ points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.