Product of sets on varieties in finite fields

TL;DR

This paper improves bounds on the product set size of subsets on varieties over finite fields, especially paraboloids, by connecting the problem to distance problems in lower dimensions.

Contribution

It demonstrates that the known threshold for product set size can be lowered for paraboloids by linking to lower-dimensional distance problems.

Findings

Breaks the $d/2$ barrier for paraboloids in certain dimensions.

Uses recent advances in distance problems to improve product set bounds.

Establishes a new connection between product sets and distance problems in finite fields.

Abstract

Let $V$ be a variety in $\mathbb{F}_q^d$ and $E\subset V$. It is known that if any line passing through the origin contains a bounded number of points from $E$, then $|\prod(E)|=|\{x\cdot y\colon x, y\in E\}|\gg q$ whenever $|E|\gg q^{\frac{d}{2}}$. In this paper, we show that the barrier $\frac{d}{2}$ can be broken when $V$ is a paraboloid in some specific dimensions. The main novelty in our approach is to link this question to the distance problem in one lower dimensional vector space, allowing us to use recent developments in this area to obtain improvements.