  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 order
characters on it. These results then can be turned back into approximative
statements about finite Abelian groups.
