Spherical higher order Fourier analysis over finite fields II: additive combinatorics for shifted ideals

Abstract

This paper is the second 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 study additive combinatorial properties for shifted modules, i.e. the structure of sets of the form $E\hat{+} E$, where $E$ is a collection of shifted modules of the polynomial ring $\mathbb{R}[x_{1},\dots,x_{d}]$ and we identify two modules if their difference contains the zero polynomial. We show that under appropriate definitions, the set $E\hat{+} E$ enjoys properties similar to the conventional setting where $E$ is a subset of an abelian group. In particular, among other results, we prove the Balog-Gowers-Szemerédi theorem, the Rusza's quasi triangle inequality and a weak form of the Plünnecke-Rusza theorem in the setting of shifted modules. We also show that for a special class of maps $ξ$ from $\mathbb{Z}_{K}^{d}$ to the collection of all shifted modules of $\mathbb{R}[x_{1},\dots,x_{d}]$, if the set $ξ(\mathbb{Z}_{K}^{d})+ξ(\mathbb{Z}_{K}^{d})$ has large additive energy, then $ξ$ is an almost linear Freiman homomorphism. This result is the crucial additive combinatorial input we need to prove the spherical Gowers inverse theorem in later parts of the series.