A quantitative inverse theorem for the $U^4$ norm over finite fields

Abstract

A remarkable result of Bergelson, Tao and Ziegler implies that if $c>0$, $k$ is a positive integer, $p\ge k$ is a prime, $n$ is sufficiently large, and $f:{\mathbb{F}}_p^n\to \mathbb{C}$ is a function with $\Vert f{\Vert }_{\infty }\le 1$ and $\Vert f{\Vert }_{U^k}\ge c$, then there is a polynomial $\pi$ of degree at most $k-1$ such that ${\mathbb{E}}_{x}f(x){\omega }^{-\pi (x)}\ge c^{\prime }$, where $\omega =\exp (2\pi i/p)$ and $c^{\prime }>0$ is a constant that depends on $c,k$ and $p$ only. A version of this result for low-characteristic was also proved by Tao and Ziegler. The proofs of these results do not yield a lower bound for $c^{\prime }$. Here we give a different proof in the high-characteristic case when $k=4$, which enables us to give an explicit estimate for $c^{\prime }$. The bound we obtain is roughly doubly exponential in the other parameters.

104 pages

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