论文标题
Freiman多塑形的逆定理
An inverse theorem for Freiman multi-homomorphisms
论文作者
论文摘要
令$ g_1，\ dots，g_k $和$ h $是有限字段$ \ mathbb {f} _p $ prime订单的矢量空间。令$ a \ subset g_1 \ times \ dots \ times g_k $为一组尺寸$δ| g_1 | \ cdots | g_k | $。令映射$ ϕ \ colon a \ to h $为多态形态，这意味着对于[k] $中的每个方向$ d \，以及每个元素$（x_1，\ dots，x_ {d-1}，x_ {d-1}，x_ {d+1} g_ {d+1} \ times \ dots \ times g_k $，发送每个$ y_d $的地图，使得$（x_1，\ dots，$ $ x_ {d-1}，$ $ $ y_d，$ $ x__ {d+1} $ x_ {D-1}，$ $ y_d，$ $ x_ {d+1}，\ dots，$ $ x_k）$是弗雷曼同构（订单2）。在本文中，我们证明，对于每张这样的地图，都有一个多芬映射$φ\ colon g_1 \ times \ dots \ times \ times g_k \ to h $，以至于一组密度上的$ ϕ =φ$ $ \ big（\ exp^{（o_k（1））}（o_ {k，p}（δ^{ - 1}））\ big）^{ - 1} $，其中$ \ exp^{（t）} $表示$ t $ - foldential。 该定理的应用包括： $ \ bullet $用于近似多项式的定量逆定理$ g $ to $ h $，用于有限维度$ \ mathbb {f} _p $ - vector-vector-vector-vector-vector-vector $ g $和$ h $，在高召唤案例中， $ \ bullet $ a量定定理，用于高示例性案例中有限领域的统一规范， $ \ bullet $ $ g_1 \ times \ dots \ times g_k $的密集子集的定量结构定理是主说明中的子空间（没有其他特征假设）。
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 $δ|G_1| \cdots |G_k|$. Let a map $ϕ\colon A \to H$ 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 $ϕ(x_1, \dots,$ $x_{d-1},$ $y_d,$ $x_{d+1}, \dots,$ $x_k)$ is a Freiman homomorphism (of order 2). In this paper, we prove that for each such map, there is a multiaffine map $Φ\colon G_1 \times\dots\times G_k \to H$ such that $ϕ= Φ$ on a set of density $\Big(\exp^{(O_k(1))}(O_{k,p}(δ^{-1}))\Big)^{-1}$, 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).
