A new perspective on the distance problem over prime fields

Abstract

Let ${\mathbb{F}}_p$ be a prime field, and $\mathcal{E}$ a set in ${\mathbb{F}}_p^2$. Let $\Delta (\mathcal{E})=\{||x-y||:x,y\in \mathcal{E}\}$, the distance set of $\mathcal{E}$. In this paper, we provide a quantitative connection between the distance set $\Delta (\mathcal{E})$ and the set of rectangles determined by points in $\mathcal{E}$. As a consequence, we obtain a new lower bound on the size of $\Delta (\mathcal{E})$ when $\mathcal{E}$ is not too large, improving a previous estimate due to Lund and Petridis and establishing an approach that should lead to significant further improvements.

References in corpus (1)

• A short proof of Rudnev's point-plane incidence bound

Cited by in corpus (1)

• Distance problems for planar hypercomplex numbers