Spherical higher order Fourier analysis over finite fields IV: an application to the Geometric Ramsey Conjecture

This paper is the fourth and the last 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 proof a conjecture of Graham on the Remsey properties for spherical configurations in the finite field setting. To be more precise, we show that for any spherical configuration $X$ of $\mathbb{F}_{p}^{d}$ of complexity at most $C$ with $d$ being sufficiently large with respect to $C$ and $\vert X\vert$, and for some prime $p$ being sufficiently large with respect to $C$, $\vert X\vert$ and $\epsilon>0$, any set $E\subseteq \mathbb{F}_{p}^{d}$ with $\vert E\vert>\epsilon p^{d}$ contains at least $\gg_{C,\epsilon,\vert X\vert}p^{(k+1)d-(k+1)k/2}$ congruent copies of $X$, where $k$ is the dimension of $\text{span}_{\mathbb{F}_{p}}(X-X)$. The novelty of our approach is that we avoid the use of harmonic analysis, and replace it by the theory of spherical higher order Fourier analysis developed in previous parts of the series.