Spherical higher order Fourier analysis over finite fields III: a spherical Gowers inverse theorem

This paper is the third part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey conjecture in the finite field setting. In this paper, we prove an inverse theorem over finite field for spherical Gowers norms, i.e. a local Gowers norm supported on a sphere. We show that if the $(s+1)$-th spherical Gowers norm of a 1-bounded function $f\colon\mathbb{F}_{p}^{d}\to \mathbb{C}$ is at least $\epsilon$ and if $d$ is sufficiently large depending only on $s$, then $f$ correlates on the sphere with a $p$-periodic $s$-step nilsequence, where the bounds for the complexity and correlation depend only on $d$ and $\epsilon$. This result will be used in later parts of the series to prove the geometric Ramsey conjecture in the finite field setting.