离散平面中点构型的插值
Interpolation of point configurations in the discrete plane
AI总结：
本文提出一种新方法，研究由两个三角形在顶点处相连的图在有限域平面上的广义距离集，在阈值 $|E|\geq q^{12/7}$ 下得到下界 $cq^6$。
摘要：
在有限域上形式化定义距离为 $||x-y||:=(x_1-y_1)^2+\cdots + (x_d-y_d)^2$，其中 $x,y\in \mathbb{F}_q^d$，距离问题自然类比于 Erdős 和 Falconer 在欧氏空间中研究的问题。给定图 $G$ 和集合 $E\subseteq \mathbb{F}_q^2$，令 $\Delta_G(E)$ 为对应于 $G$ 的广义距离集。当 $G$ 是 $k+1$ 个顶点上的完全图时，Bennett、Hart、Iosevich、Pakianathan 和 Rudnev 证明了当 $|E|\geq q^{d-\frac{d-1}{k+1}}$ 时，有 $|\Delta_G(E)|\geq cq^{\binom{k+1}{2}}$。当 $k=d=2$ 时，阈值可改进为 $|E|\geq q^{\frac{8}{5}}$。此外，Jardine、Iosevich 和 McDonald 证明了当 $G$ 是 $k+1$ 个顶点上的树时，只要 $E\subseteq \mathbb{F}_q^d$，$d\geq 2$ 满足 $|E|\geq C_kq^{\frac{d+1}{2}}$，则 $\Delta_G(E)=\mathbb{F}_q^k$。本文提出一种技术，使我们能够研究同时具有刚性组件和非刚性组件的某些图。特别地，我们证明对于 $E\subseteq \mathbb{F}_q^2$，$q=p^n$，$n$ 为奇数，$p\equiv 3 \\ \text{mod} \\ 4$，且 $G$ 是由两个三角形在顶点处相连构成的图，则当 $|E|\geq q^{\frac{12}{7}}$ 时，有 $|\Delta_G(E)|\geq cq^6$。
英文摘要：
Defining distances over finite fields formally by $||x-y||:=(x_1-y_1)^2+\cdots + (x_d-y_d)^2$ for $x,y\in \mathbb{F}_q^d$, distance problems naturally arise in analogy to those studied by Erdős and Falconer in Euclidean space. Given a graph $G$ and a set $E\subseteq \mathbb{F}_q^2$, let $Δ_G(E)$ be the generalized distance set corresponding to $G$. In the case when $G$ is the complete graph on $k+1$ vertices, Bennett, Hart, Iosevich, Pakianathan, and Rudnev showed that when $|E|\geq q^{d-\frac{d-1}{k+1}}$, it follows that $|Δ_G(E)|\geq cq^{\binom{k+1}{2}}$. In the case when $k=d=2$, the threshold can be improved to $|E|\geq q^{\frac{8}{5}}$. Moreover, Jardine, Iosevich, and McDonald showed that in the case when $G$ is a tree with $k+1$ vertices, then whenever $E\subseteq \mathbb{F}_q^d$, $d\geq 2$ satisfies $|E|\geq C_kq^{\frac{d+1}{2}}$, it follows that $Δ_G(E)=\mathbb{F}_q^k$. In this paper, we present a technique which enables us to study certain graphs with both rigid and non-rigid components. In particular, we show that for $E\subseteq \mathbb{F}_q^2$, $q=p^n$, $n$ odd, $p\equiv 3 \ \text{mod} \ 4$, and $G$ is the graph consisting of two triangles joined at a vertex, then whenever $|E|\geq q^{\frac{12}{7}}$, it follows that $|Δ_G(E)|\geq cq^6$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
