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

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

2010-11-20

arxiv.org/abs/1011.4600v3

Mathematics

Number Theory

final version, 25 pages

Scientific paper

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 $\|f_2\|_{U^k}$ 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 $\mathbb{F}_p^n$.

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