Pinned algebraic distances determined by Cartesian products in   $\mathbb{F}_p^2$
  Let $p$ be an odd prime and $A \subseteq \mathbb{F}_p$ be a subset of the finite field with $p$ elements. We show that $A \times A \subseteq \mathbb{F}_p^2$ determines at least a constant multiple of $\min\{p, |A|^{3/2}\}$ distinct pinned algebraic distances. 
Giorgis Petridis
math.CO