Parallelepipeds, nilpotent groups and Gowers norms

Abstract

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 3 and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups. Résumé (Parallélépipèdes, groupes nilpotents et normes de Gowers)Dans sa preuve du théorème de Szemerédi, Gowers a introduit certaines normes définies par sommation sur des parallélépipèdes. Il est naturel de se demander sous quelles hypothèses on peut généraliser sa définition des parallélépipèdes et donc de ses normes. Nous nous restreignons aux dimensions 2 et 3 et décrivons une correspondance entre structures de parallélépipèdes et groupes nilpotents.

Higher Order Fourier Analysis

“…Important results on the fifth and sixth topics were also obtained by Host and Kra in the papers [13], [14]. The paper [13] is the main motivation of [1] which is the corner stone of our approach.…”

“…The nilspace axiom system is a variant of the Host-Kra axiom system for parallelepiped structures [13]. In [13] the two step case is analyzed and it is proved that the structures are tied to two nilpotent groups.…”

Graph norms and Sidorenko’s conjecture

“…In Step 1, for every v ∈ Q n−1 , we grouped the two vertices 0v, 1v and the edge between them as one vertex (see (19)) and this reduced Q n to Q n−1 . In this step we reduce Q n to Q 2 .…”

“…Recently Gowers [13,14] defined a hypergraph version of this norm, and subsequently he [12] and Nagle, Rödl, Schacht, and Skokan [22,26,25] independently established a hypergraph regularity lemma which easily implies Szemerédi's theorem in its full generality, and even stronger theorems such as Furstenberg-Katznelson's multi-dimensional arithmetic progression theorem [24,9], a result that the only known proof for it at the time was through ergodic theory [11]. In fact arithmetic version of the Gowers norm has interesting interpretations in ergodic theory, and has been studied from that aspect [19]. The discovery of this norm led to a better understanding of the concept of quasirandomness, and provided strong tools.…”

Untitled

“…In a series of papers [8,51,52], one joint with Omar Camarena, he works with abstract structures that he calls nilspaces, which are variants of abstract parallelepiped structures introduced by Host and Kra [39] (abstracting out certain arguments from [38]). Such structures can be thought of as the most general structures for which one can make sense of uniformity norms, and are therefore a natural setting for thinking about inverse theorems.…”