Higher order Fourier analysis as an algebraic theory I

Abstract

Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemerédi type theorems in abelian groups. In this paper we develop an algebraic theory that creates a connection between these approaches. Our main method is to take the ultra product of abelian groups and to develop a precise algebraic theory of higher order characters on it. These results then can be turned back into approximative statements about finite Abelian groups.

Cited by in corpus (6)

• Nilspaces, nilmanifolds and their morphisms
• On higher order Fourier analysis
• Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors
• Gowers norms, regularization and limits of functions on abelian groups
• An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement)
• Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups