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 $A\subset G_1\times \cdots \times G_k$ be a set of size $\delta \mathrm{\mid }{G}_1\mathrm{\mid }\cdots \mathrm{\mid }{G}_k\mathrm{\mid }$. Let a map $\varphi \text{\hspace{0.17em}⁣}:A\to H$ 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 $G_1\times\dots\times G_{d-1}\times G_{d+1}\times \dots\times G_k$,themapthatsendseach$y_d$suchthat$(x_1, \dots,$ $x_{d-1},$ $y_d,$ $x_{d+1},\dots ,$ $x_k)\in A$ to $\varphi (x_1,\dots ,$ $x_{d-1},$ $y_d,$ $x_{d+1},\dots ,$ $x_k)$ is a Freiman homomorphism (of order 2). In this paper, we prove that for each such map, there is a multiaffine map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:G_1\times \cdots \times G_k\to H$ such that $\varphi =\mathrm{\Phi }$ on a set of density $({\exp }^{(O_k(1))}(O_{k,p}({\delta }^{-1})){)}^{-1}$, 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).