General inverse theory for the $\mathsf{U}^4$ norm

TL;DR

This paper develops a quantitative inverse theory for the Gowers U^4 norm in finite abelian groups, introducing almost-cubic polynomials as obstructions to uniformity and establishing quasipolynomial inverse theorems.

Contribution

It introduces almost-cubic polynomials as new obstructions and proves quasipolynomial inverse theorems for the U^4 norm in general finite abelian groups.

Findings

Identification of almost-cubic polynomials as obstructions to uniformity

Quasipolynomial inverse theorem for groups with (|G|,6)=1

Existence of cubic polynomials for groups of the form (Z/2^dZ)^n

Abstract

In this paper, we develop a quantitative inverse theory for the Gowers uniformity norm $\|\cdot\|_{\mathsf{U}^4}$ in general finite abelian groups. We identify a new type of obstructions to uniformity, which we call almost-cubic polynomials. An almost-cubic polynomial $q$ on a Bohr set $B(\Gamma, \rho_0)$ is a function such that, for each $\rho \leq \min\{\rho_0, 1/8\}$, we have \[\|\Delta_{a,b,c,d} q(x)\|_{\mathbb{T}} \leq 2^{10} \rho\] for all $x, a,b,c,d \in B(\Gamma, \rho)$. Let $f : G \to \mathbb{D}$ be a function with $\|f\|_{\mathsf{U}^4} \geq c$. We prove quasipolynomial inverse theorems: $\bullet$ when $(|G|, 6) = 1$, there exists an almost-cubic $q : B(\Gamma, \rho)$ for $|\Gamma| \leq \log^{O(1)} c^{-1}$ and $\rho \geq \exp(-\log^{O(1)} c^{-1})$, and an element $t \in G$ such that $$\Big|\sum_{x \in G} 1_{B}(x) f(x + t) \operatorname{e}(q(x))\Big| \geq \exp(-\log^{O(1)} c^{-1})|G|,$$ $\bullet$ when $G = (\mathbb{Z}/2^d\mathbb{Z})^n$, there exists a cubic polynomial $q : G \to \mathbb{T}$ such that $$\Big|\sum_{x \in G} f(x)\operatorname{e}(q(x))\Big| \geq \exp(-\log^{O_d(1)} c^{-1})|G|.$$ Almost-cubic polynomials are rather rigid and we exhibit a strong connection with generalized polynomials in the case of cyclic groups, as well as with polynomials in the classical sense in the case of finite vector spaces. We also answer a question of Jamneshan, Shalom and Tao concerning the inverse theory in groups of bounded torsion. The central result from which the inverse theorems follow is a structural result for Freiman bihomomorphisms in general finite abelian groups. In our proof, we generalize methods of our previous work in the case of finite vector spaces and introduce novel ideas concerning extensions of Freiman bihomomorphisms. In the problem of extension of Freiman bihomomorphisms, genuinely new phenomena appear in general finite abelian groups.