论文标题
注意有限领域的固定距离问题
Note on the pinned distance problem over finite fields
论文作者
论文摘要
令F_Q为具有奇数元素的有限字段。在本文中，我们证明，如果e \ subseteq \ mathbb f_q^d，d \ ge 2和| e | \ ge q q，则存在一个集合y \ subseteq \ subseteq \ mathbb f_q^d，with | y | y | y | y | y | y | y | y | y | y | y | q^d $在y中的数量和范围fin fin fin和fin的数字相似。作为推论，我们可以在| e | e | \ ge q上获得每组E \ subseteq \ mathbb f_q^d，存在一个y \ subseteq \ subseteq \ subseteq \ mathbb f_q^d with | y | \ sim q^d，以便任何设置e \ cup \ \ \ \ \ y \ y y y y \ y y y \ y y \ y y \ y y \ y y \ y h均确定为y \ y \。平均论点和鸽子原理在证明我们的结果方面起着至关重要的作用。
Let F_q be a finite field with odd q elements. In this article, we prove that if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y \subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number of distances between the point y and the set E is similar to the size of the finite field \mathbb F_q. As a corollary, we obtain that for each set E\subseteq \mathbb F_q^d with |E|\ge q, there exists a set Y\subseteq \mathbb F_q^d with |Y|\sim q^d so that any set E\cup \{y\} with y\in Y determines a positive proportion of all possible distances. An averaging argument and the pigeonhole principle play a crucial role in proving our results.
