A new perspective on the distance problem over prime fields

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.