Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

Abstract

We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host-Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host-Kra type seminorms for measure-preserving actions of countable nilpotent groups.This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability.These are sequences indexed by the infinite discrete cube, such that for every integer k ≥ 0 the joint distribution's marginals on affine subcubes of dimension k are all equal.In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

“…As detailed below, we deduce Theorem 1.5 from results on cubic couplings from [6]. In particular, this yields directly that the nilspace polynomial in this result is arbitrarily well balanced in relation to its complexity (this then holds also in the inverse theorem; see Theorem 5.2).…”

“…We argue by contradiction, supposing that there is a sequence of 1-bounded Borel functions f i : X i → C that disproves the theorem (thus for some ǫ > 0 and real numbers N i → ∞ as i → ∞, for each i the required decomposition fails for f i , ǫ and N i ). We then consider the 1-bounded function f = lim ω f i : X → C, and analyze this using results on cubic couplings from [6]. To detail this further, we need to recall the definition of a cubic coupling, and for this purpose we first have to recall the following notation from [6].…”

“…We then consider the 1-bounded function f = lim ω f i : X → C, and analyze this using results on cubic couplings from [6]. To detail this further, we need to recall the definition of a cubic coupling, and for this purpose we first have to recall the following notation from [6].…”

“…17]. In [6], a framework for such a unification was put forward, based on the concept of a cubic coupling, and a first application was given by recovering and generalizing the ergodic structure theorem in this framework. Another application was announced in that paper, namely a result extending the inverse theorem 2 PABLO CANDELA AND BAL ÁZS SZEGEDY from [16] to compact abelian groups and also to nilmanifolds and more general nilspaces.…”

“…In Section 3, we analyze ultraproducts of cfr coset nilspaces to locate certain factors that have a cubic coupling structure. This will enable us to apply a structure theorem from [6], as a crucial step in our proof of Theorem 1.5. In Section 4, we prove a new stability result for morphisms into cfr nilspaces, Theorem 4.2, which is a central element in our proof of Theorem 1.5 and seems to be also of intrinsic interest.…”