Higher-Order Fourier Analysis Of {\mathbb{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 \mathbbF_pⁿ{\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 ||f₂||_U^k{\|{f_2}\|_{U^k}}, is small for a proper value of k.