Gowers norms, regularization and limits of functions on abelian groups

Abstract

For every natural number k we prove a decomposition theorem for bounded measurable functions on compact abelian groups into a structured part, a quasi random part and a small error term. In this theorem quasi randomness is measured with the Gowers norm U(k+1) and the structured part is a bounded complexity ``nilspace-polynomial'' of degree k. This statement implies a general inverse theorem for the U(k+1) norm. (We discuss some consequences in special families of groups such as bounded exponent groups, zero characteristic groups and the circle group.) Along these lines we introduce a convergence notion and corresponding limit objects for functions on abelian groups. This subject is closely related to the recently developed graph and hypergraph limit theory. An important goal of this paper is to put forward a new algebraic aspect of the notion ``higher order Fourier analysis''. According to this, k-th order Fourier analysis is regarded as the study of continuous morphisms between structures called compact k-step nilspaces. All our proofs are based on an underlying theory of topological nilspace factors of ultra product groups.

References in corpus (7)

• The true complexity of a system of linear equations
• Linear forms and higher-degree uniformity for functions on ${\mathbb{F}}_p^n$
• Nilspaces, nilmanifolds and their morphisms
• A measure-theoretic approach to the theory of dense hypergraphs
• Higher order Fourier analysis as an algebraic theory I
• A point of view on Gowers uniformity norms
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Cited by in corpus (7)

• Nilspaces, nilmanifolds and their morphisms
• On higher order Fourier analysis
• Large gaps between consecutive prime numbers
• Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups
• Notes on nilspaces: algebraic aspects
• The inverse conjecture for the Gowers norm over finite fields in low characteristic
• Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations