Higher-order Fourier analysis of ${\mathbb{F}}_p^n$ and the complexity of systems of linear forms

Abstract

Consider a subset $A$ of ${\mathbb{F}}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degree of uniformity $k$ such that assuming that $\Vert f_2{\Vert }_{U^k}$ is sufficiently small, it is possible to discard $f_2$ and replace $1_A$ with $f_1$ in the average over this family of linear forms, affecting it 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. Linear forms and higher-degree uniformity for functions on ${\mathbb{F}}_p^n$. Geom. Funct. Anal., 21(1):36--69, 2011] regarding the analytic averages that involve more than one subset of ${\mathbb{F}}_p^n$.] regarding the analytic averages that involve more than one subset of ${\mathbb{F}}_p^n$.

final version, 25 pages

References in corpus (1)

• Worst Case to Average Case Reductions for Polynomials