固定距离集、有限域中的Wolff指数及改进的和积估计
Pinned distance sets, Wolff's exponent in finite fields and improved sum-product estimates
摘要：
在有限域上的向量空间中的Falconer距离问题的一个类似物要求阈值$α>0$，使得当$|E| \gtrsim q^α$时，$|Δ(E)| \gtrsim q$，其中$E \subset {\Bbb F}_q^d$，有限域上的$d$维向量空间，具有$q$元素（不一定是素数）。这里是$Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$。第二个列出的作者和Misha Rudnev建立了阈值$\frac{d+1}{2}$，本文的作者Doowon Koh和Misha Rudnev证明了这个指数在偶数维中是尖锐的。本文在$E$具有乘积结构的假设下，将阈值改进为$\frac{d^2}{2d-1}$。特别地，我们得到了4/3的指数，与Wolff在欧氏空间中得到的相应指数一致。
英文摘要：
An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $α>0$ such that $|Δ(E)| \gtrsim q$ whenever $|E| \gtrsim q^α$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. The second listed author and Misha Rudnev established the threshold $\frac{d+1}{2}$, and the authors of this paper, Doowon Koh and Misha Rudnev proved that this exponent is sharp in even dimensions. In this paper we improve the threshold to $\frac{d^2}{2d-1}$ under the additional assumption that $E$ has product structure. In particular, we obtain the exponent 4/3, consistent with the corresponding exponent in Euclidean space obtained by Wolff.
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