Higher-Order Fourier Analysis Of 𝔽_p^n And The Complexity Of Systems Of Linear Forms

In this article we are interested in 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 pseudo-random in the sense that the k -th Gowers uniformity norm of f 2 , denoted by 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 main result solves Problem 7.6 in W.T. Gowers and J. Wolf’s paper [GW2], regarding the analytic averages that involve more than one subset of 𝔽_p^n .