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

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 $B(Γ, ρ_0)$ is a function such that, for each $ρ\leq \min\{ρ_0, 1/8\}$, we have \[\|Δ_{a,b,c,d} q(x)\|_{\mathbb{T}} \leq 2^{10} ρ\] for all $x, a,b,c,d \in B(Γ, ρ)$. 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 $q : B(Γ, ρ)$ for $|Γ| \leq \log^{O(1)} c^{-1}$ and $ρ\geq \exp(-\log^{O(1)} c^{-1})$, 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.