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

TL;DR

This paper improves the threshold for the finite field Falconer distance problem for sets with product structure, establishing a new exponent of 4/3 that aligns with Euclidean space results.

Contribution

The authors improve the distance set threshold in finite fields for product-structured sets, matching Euclidean exponents and advancing understanding of geometric configurations.

Findings

Established a new threshold exponent of 4/3 for product sets in finite fields.

Proved the sharpness of the threshold in even dimensions.

Extended the analogy between finite field and Euclidean distance problems.

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 $E \subset {\Bbb F}_q^d$, 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.