Abstract:In a famous paper Timothy Gowers introduced a sequence of
norms U(k) defined for functions on abelian groups. He used these
norms to give quantitative bounds for Szemeredi's theorem on
arithmetic progressions. The behavior of the U(2) norm is closely tied to Fourier analysis. In
this talk we present a generalization of Fourier analysis (called k-th
order Fourier analysis) that is related in a similar way to the U(k+1)
Ordinary Fourier analysis deals with homomorphisms of abelian groups
into the circle group. We view k-th order Fourier analysis as a theory which deals with
morphisms of abelian groups into algebraic structures that we call
"k-step nilspaces". These structeres are variants of structures
introduced by Host and Kra (called parallelepiped structures) and they
are close relatives of nil-manifolds. Our approach has two components.
One is an uderlying algebraic theory of nilspaces and the other is a
variant of ergodic theory on ultra product groups.
Using this theory, we obtain inverse theorems for the U(k) norms on
arbitrary abelian groups that generalize results by Green, Tao and
Ziegler. As a byproduct we also obtain an interesting limit theory for
functions on abelian groups in the spirit of the recently developed
graph limit theory.