Product of sets on varieties 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.