Higher-order Fourier analysis of F_p^n and the complexity of systems of
linear forms

We study the density of small linear structures (e.g.
arithmetic progressions) in subsets A of the group F_p^n. It is possible
to express these densities as certain analytic averages involving 1_A,
the indicator function of A. In the higher-order Fourier analytic
approach, the function 1_A is decomposed as a sum f_1+f_2 where f_1 is
structured in the sense that it has a simple higher-order Fourier
expansion, and f_2 is pseudorandom in the sense that the kth Gowers
uniformity norm of f_2, dentoted \|f_2\|_{U^k}, is small for a proper
value of k.

For a given linear structure, we find the smallest degree of uniformity
k such that assuming that | f_2 |_{U^k} is sufficiently small, it is
possible to discard f_2 and replace 1_A with f_1, affecting the
corresponding analytic average only negligibly. Previously, Gowers and
Wolf solved this problem for the case where f_1 is a constant function.
Furthermore, our result extends to analytic averages that involve more
than one subset of F_p^n, and resolves an open problem posed by Gowers
and Wolf.

Joint work with Hamed Hatami.