Pinned distance sets, Wolff's exponent in finite fields and improved sum-product estimates

Abstract

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha >0$ such that $|\Delta (E)|\gtrsim q$ whenever $|E|\gtrsim q^{\alpha }$, where $E\subset {\mathbb{F}}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\Delta (E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2:x,y\in E\}$. 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.

Cited by in corpus (1)

• The sum-product phenomenon in arbitrary rings