  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.
