On higher order Fourier analysis

Abstract

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the notion "higher order Fourier analysis" in terms of continuous morphisms between structures called compact $k$-step nilspaces. As a byproduct of our results we obtain a new type of limit theory for functions on abelian groups in the spirit of the so-called graph limit theory. Our proofs are based on an exact (non-approximative) version of higher order Fourier analysis which appears on ultra product groups.

arXiv admin note: substantial text overlap with arXiv:1010.6211

References in corpus (9)

• 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
• Gowers norms, regularization and limits of functions on abelian groups
• A measure-theoretic approach to the theory of dense hypergraphs
• Higher order Fourier analysis as an algebraic theory I
• Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups
• An inverse theorem for the Gowers U^{s+1}[N]-norm
• A point of view on Gowers uniformity norms

Cited by in corpus (16)

• Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors
• Approximate algebraic structure
• Notes on nilspaces: algebraic aspects
• An inverse theorem for Freiman multi-homomorphisms
• Uniformity of multiplicative functions and partition regularity of some quadratic equations
• Higher order regionally proximal equivalence relations for general minimal group actions
• Notes on compact nilspaces
• Sarnak's Conjecture for nilsequences on arbitrary number fields and applications
• Growth in groups: ideas and perspectives
• Periodic nilsequences and inverse theorems on cyclic groups
• A structure theorem for multiplicative functions over the Gaussian integers and applications
• Generalizations of Fourier analysis, and how to apply them
• On the structure theory of cubespace fibrations
• The automorphism group of a graphon
• Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences
• Multiple recurrence in quasirandom groups