  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.
