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

# 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 $f_2$, denoted $|f_2\|_{U^k}$, 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.