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

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.

19 pages; v2: minor changes following referee's report

References in corpus (1)

• Singular Brascamp-Lieb inequalities with cubical structure

Cited by in corpus (5)

• A Szemerédi-type theorem for subsets of the unit cube
• Singular Brascamp-Lieb inequalities with cubical structure
• Density theorems for anisotropic point configurations
• The density of sets containing large similar copies of finite sets
• Large dilates of hypercube graphs in the plane