An Inverse Theorem for Certain Directional Gowers Uniformity Norms

TL;DR

This paper proves an inverse theorem for certain directional Gowers uniformity norms over finite vector spaces, characterizing functions with large norms via structured polynomial phases and auxiliary functions.

Contribution

It establishes the inverse theorem for a specific case of directional Gowers norms involving direct sum decompositions, extending understanding of uniformity norms in additive combinatorics.

Findings

Characterization of functions with large directional Gowers norms via polynomial phases

Development of an approximation theorem for cuboid-counting functions

Application of inverse theorems for Freiman multi-homomorphisms

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 $f \colon G \to \mathbb{C}$ 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 $\partial_u f(x) \colon= f(x + u) \overline{f(x)}$ is the discrete multiplicative derivative. Suppose that $G$ is a direct sum of subspaces $G = U_1 \oplus U_2 \oplus \dots \oplus U_k$. 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 $\|\cdot\|_{\mathsf{U}}$ for the norm above, we show that if $f \colon G \to \mathbb{C}$ is a function bounded by 1 in magnitude and obeying $\|f\|_{\mathsf{U}} \geq c$, provided $\ell < p$, one can find a polynomial $\alpha \colon G \to \mathbb{F}_p$ of degree at most $k + \ell - 1$ and functions $g_i \colon \oplus_{j \in [k] \setminus \{i\}} G_j \to \{z \in \mathbb{C} \colon |z| \leq 1\}$ for $i \in [k]$ such that \[\Big|\mathbb{E}_{x \in G} f(x) \omega^{\alpha(x)} \prod_{i \in [k]} g_i(x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_k)\Big| \geq \Big(\exp^{(O_{p,k,\ell}(1))}(O_{p,k,\ell}(c^{-1}))\Big)^{-1}.\] The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.