Parallelepipeds, Nilpotent Groups, and Gowers Norms

In his proof of Szemeredi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions 2 and 3 and show when this possible, and describe a correspondence between the parallelepiped structures nilpotent groups.