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, $p\geq k$ is a prime, $n$ is sufficiently large, and $f:\mathbb F_p^n\to\mathbb C$ is a function with $\|f\|_\infty\leq 1$ and $\|f\|_{U^k}\geq c$, then there is a polynomial $\pi$ of degree at most $k-1$ such that $\mathbb E_xf(x)\omega^{-\pi(x)}\geq c'$, where $\omega=\exp(2\pi i/p)$ and $c'>0$ 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 $c'$. Here we give a different proof in the high-characteristic case when $k=4$, which enables us to give an explicit estimate for $c'$. The bound we obtain is roughly doubly exponential in the other parameters.