标题： 低复杂度逼近：由仿射条件推广定义的集合 显示英文标题

标题： Low-complexity approximations for sets defined by generalizations of affine conditions

摘要： 设 $p$ 为一个素数，$S$ 是 $\mathbb{F}_p$ 的非空子集，并令 $0<\epsilon\leq 1$。 我们证明了存在一个常数 $C=C(p, \epsilon)$，使得对于每个正整数 $k$，只要 $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$是线性形式且 $E_1, \dots, E_k$是 $\mathbb{F}_p$的子集，则存在线性形式 $\psi_1, \dots, \psi_C: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$和子集 $F_1, \dots, F_C$ of $\mathbb{F}_p$，使得集合 $U=\{x \in S^n: \psi_1(x) \in F_1, \dots, \psi_C(x) \in F_C\}$包含于集合 $V=\{x \in S^n: \phi_1(x) \in E_1, \dots, \phi_k(x) \in E_k\}$，并且差集 $V \setminus U$在 $S^n$中的密度至多为 $\epsilon$。 我们接着将这一结果推广到另一种情形，其中$\phi_1, \dots, \phi_k$被有限阿贝尔群$G$和$H$的同态映射$G^n \to H$替换，并且推广到另一种情形，其中它们被低次的多项式映射$\mathbb{F}_p^n \to \mathbb{F}_p$替换。 显示英文摘要

摘要： Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$ such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ are linear forms and $E_1, \dots, E_k$ are subsets of $\mathbb{F}_p$, there exist linear forms $\psi_1, \dots, \psi_C: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ and subsets $F_1, \dots, F_C$ of $\mathbb{F}_p$ such that the set $U=\{x \in S^n: \psi_1(x) \in F_1, \dots, \psi_C(x) \in F_C\}$ is contained inside the set $V=\{x \in S^n: \phi_1(x) \in E_1, \dots, \phi_k(x) \in E_k\}$, and the difference $V \setminus U$ has density at most $\epsilon$ inside $S^n$. We then generalize this result to one where $\phi_1, \dots, \phi_k$ are replaced by homomorphisms $G^n \to H$ for some pair of finite Abelian groups $G$ and $H$, and to another where they are replaced by polynomial maps $\mathbb{F}_p^n \to \mathbb{F}_p$ of small degree.