Higher order Fourier analysis as an algebraic theory III

For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have k-nilpotent factor groups explaining why k-th order Fourier analysis has non-commutative features. To demonstrate our method in the quadratic case we develop a new quadratic representation theory on finite abelian groups. We introduce the notion of a quadrtic nil-morphism of an abelian group into a two step nil-manifold. We prove a structure theorem saying that any bounded function on a finite abelian group is decomposable into a structured part (which is the composition of a nil-morphism with a bounded complexity continuous function) and a random looking part with small U3 norm. It implies a new inverse theorem for the U3 norm. (The general case for Un, n>3 will be discussed in the next part of this sequence.) We point out that our framework creates interesting limit objects for functions on finite (or compact) abelian groups that are measurable functions on nil-manifolds.