Higher-Order Fourier Analysis Of \({\mathbb{F}_{p}^n}\) And The Complexity Of Systems Of Linear Forms

Abstract In this article we are interested in the density of small linear structures (e.g. arithmetic progressions) in subsets A of the group \({\mathbb{F}_{p}^n}\) . 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 \({\mathbb{F}_{p}^n}\).