An inverse theorem for Freiman multi-homomorphisms

Let $G_1, \dots, G_k$ and $H$ be vector spaces over a finite field $\mathbb{F}_p$ of prime order. Let $A \subset G_1 \times\dots\times G_k$ be a set of size $\delta |G_1| \cdots |G_k|$. Let a map $\phi \colon A \to H$ be a multi-homomorphism, meaning that for each direction $d \in [k]$, and each element $(x_1, \dots, x_{d-1}, x_{d+1}, \dots, x_k)$ of $G_1\times\dots\times G_{d-1}\times G_{d+1}\times \dots\times G_k$, the map that sends each $y_d$ such that $(x_1, \dots,$ $x_{d-1},$ $y_d,$ $x_{d+1}, \dots,$ $x_k) \in A$ to $\phi(x_1, \dots,$ $x_{d-1},$ $y_d,$ $x_{d+1}, \dots,$ $x_k)$ is a Freiman homomorphism (of order 2). In this paper, we prove that for each such map, there is a multiaffine map $\Phi \colon G_1 \times\dots\times G_k \to H$ such that $\phi = \Phi$ on a set of density $\Big(\exp^{(O_k(1))}(O_{k,p}(\delta^{-1}))\Big)^{-1}$, where $\exp^{(t)}$ denotes the $t$-fold exponential. Applications of this theorem include: $\bullet$ a quantitative inverse theorem for approximate polynomials mapping $G$ to $H$, for finite-dimensional $\mathbb{F}_p$-vector spaces $G$ and $H$, in the high-characteristic case, $\bullet$ a quantitative inverse theorem for uniformity norms over finite fields in the high-characteristic case, and $\bullet$ a quantitative structure theorem for dense subsets of $G_1 \times\dots\times G_k$ that are subspaces in the principal directions (without additional characteristic assumptions).