  Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space
over a finite field with $q$ elements, which satisfies $|E| > q$. There exist
$x,y \in E$ such that $|E \cdot (y-x)| > q/2.$ In particular, $(E+E) \cdot
(E-E) = \mathbb{F}_q$.
