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

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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_{U^k}\ge c$, then there is a polynomial $\pi$ of degree at most $k-1$ such that ${\mathbb{E}}_{x}f(x){\omega }^{-\pi (x)}\ge c^{\mathrm{\prime }}$, where $\omega=\exp(2\pi i/p)$and$c’>0$isaconstantthatdependson$c,k$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 $c^{\mathrm{\prime }}$. Here we give a different proof in the high-characteristic case when $k=4$, which enables us to give an explicit estimate for $c^{\mathrm{\prime }}$. The bound we obtain is roughly doubly exponential in the other parameters.