We study the Erd?s–Falconer distance problem for a set A ? F 2 $A\subset \mathbb {F}^2$ , where $\mathbb {F}$ is field of positive characteristic p $p$ . If = {F}=\mathbb {F}_p$ and cardinality | $|A|$ exceeds 5 / 4 $p^{5/4}$ we prove that $A$ determines an asymptotically full proportion feasible distances. For small sets namely when ? 3 $|A|\leqslant p^{4/3}$ over any either ? $\gg |A|^{2/3}$ ...
