An Inverse Theorem for Certain Directional Gowers Uniformity Norms

Abstract

Let $G$ be a finite-dimensional vector space over a prime field ${\mathbb{F}}_p$ with some subspaces $H_1,\dots ,H_k$. Let $f:G\to \mathbb{C}$ be a function. Generalizing the notion of Gowers uniformity norms, Austin introduced directional Gowers uniformity norms of $f$ over $(H_1,\dots ,H_k)$ as \[\|f\|_{\mathsf{U}(H_1, \dots, H_k)}^{2^k} = \mathbb{E}_{x \in G,h_1 \in H_1, \dots, h_k \in H_k} \partial_{h_1} \dots \partial_{h_k} f(x)\] where ${\partial }_{u}f(x):=f(x+u)\overline{f(x)}$ is the discrete multiplicative derivative. Suppose that $G$ is a direct sum of subspaces $G=U_1\oplus U_2\oplus \cdots \oplus U_k$. In this paper we prove the inverse theorem for the norm \[\|\cdot\|_{\mathsf{U}(U_1, \dots, U_k, \smash[b]{\underbrace{\scriptstyle G, \dots, G}_{\scriptscriptstyle \ell}})},\] which is the simplest interesting unknown case of the inverse problem for the directional Gowers uniformity norms. Namely, writing $\Vert \cdot {\Vert }_{\mathsf{U}}$ for the norm above, we show that if $f:G\to \mathbb{C}$ is a function bounded by 1 in magnitude and obeying $\Vert f{\Vert }_{\mathsf{U}}\ge c$, provided $\ell <p$, one can find a polynomial $\alpha :G\to {\mathbb{F}}_p$ of degree at most $k+\ell -1$ and functions $g_i:{\oplus }_{j\in [k]\setminus \{i\}}{G}_j\to \{z\in \mathbb{C}:|z|\le 1\}$ for $i\in [k]$ such that \[\Big|\mathbb{E}_{x \in G} f(x) ω^{α(x)} \prod_{i \in [k]} g_i(x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_k)\Big| \geq \Big(\exp^{(O_{p,k,\ell}(1))}(O_{p,k,\ell}(c^{-1}))\Big)^{-1}.\] The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.

57 pages