Inverse Theorem for Certain Directional Gowers Uniformity Norms

Luka Milićević

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 \[ \big\|f\big\|_{\mathsf{U}(H_1,\dots,H_k)}^{2^k}=\E_{xı G,h_1ı H_1,\dots,h_kı H_k}\mder_{h_1}\dots\mder_{h_k} f(x) \] where $\mder_u f(x)\colon=f(x+u)\overline{f(x)}$ 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 \[ \|\cdot\|_{\mathsf{U}(U_1,\dots,U_k,G,\dots,G)}, \] with $\ell$ copies of $G$ in the subscript, 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 $g_i\colon\oplus_{j\in [k]\setminus\{i\}} U_j\to\{z\in\mathbb{C}\colon |z|\leq 1\}$ for $i\in [k]$ such that \begin{align*} |\E_{xı G} f(x)mega^{lpha(x)}rod_{iı [k]} g_i(x_1,\dots,&x_{i-1},x_{i+1},\dots,x_k)| [-1ex] &\geq\big(\exp^{(O_{p,k,\ell}(1))}(O_{p,k,\ell}({c}^{-1}))\big)^{-1}. \end{align*} The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.