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.