Parallelepipeds, nilpotent groups and Gowers norms

Abstract

In his proof of Szemerédi'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 3and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups. ©SMF.