Nilspaces, nilmanifolds and their morphisms

Abstract

Recent developments in ergodic theory, additive combinatorics, higher order Fourier analysis and number theory give a central role to a class of algebraic structures called nilmanifolds. In the present paper we continue a program started by Host and Kra. We introduce nilspaces as structures satisfying a variant of the Host-Kra axiom system for parallelepiped structures. We give a detailed structural analysis of abstract and compact topological nilspaces. Among various results it will be proved that compact nilspaces are inverse limits of finite dimensional ones. Then we show that finite dimensional compact connected nilspaces are nilmanifolds. The theory of compact nilspaces is a generalization of the theory of compact abelian groups. This paper is the main algebraic tool in the second authors approach to Gowers's uniformity norms and higher order Fourier analysis.

References in corpus (5)

• 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

Cited by in corpus (22)

• On higher order Fourier analysis
• Some open problems on multiple ergodic averages
• Strictly ergodic distal models and a new approach to the Host-Kra factors
• Gowers norms, regularization and limits of functions on abelian groups
• Weighted multiple ergodic averages and correlation sequences
• Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups
• A point of view on Gowers uniformity norms
• Notes on nilspaces: algebraic aspects
• On nilspace systems and their morphisms
• Ergodic theorems for polynomials in nilpotent groups
• The inverse conjecture for the Gowers norm over finite fields in low characteristic
• Host-Kra theory for ${⨁}_{p\in P}{\mathbb{F}}_p$- Systems and multiple recurrence
• Notes on compact nilspaces
• Periodic nilsequences and inverse theorems on cyclic groups
• Combinatorial properties of Nil-Bohr sets
• Large values of the Gowers-Host-Kra seminorms
• Generalizations of Fourier analysis, and how to apply them
• Quantitative inverse theory of Gowers uniformity norms
• On the structure theory of cubespace fibrations
• Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences
• Directional dynamical cubes for minimal ${\mathbb{Z}}^d$-systems
• ${⨁}_{p\in P}{\mathbb{F}}_p$-Systems as Abramov Systems