201203.2260v1 On higher order Fourier analysis [关于高阶傅里叶分析]

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers´s uniformity norms. We put forward an algebraic interpretation of the notion "higher order Fourier analysis" in terms of continuous morphisms between structures called compact k-step nilspaces. ..