Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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).

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The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci