200903.0897v1 Higher order Fourier analysis as an algebraic theory I

Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemer\´edi 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 ord..