On the pinned distances problem in positive characteristic

Abstract

We study the Erdős–Falconer distance problem for a set A ⊂ F 2 $A\subset \mathbb {F}^2$ , where F $\mathbb {F}$ is a field of positive characteristic p $p$ . If F = F p $\mathbb {F}=\mathbb {F}_p$ and the cardinality | A | $|A|$ exceeds p 5 / 4 $p^{5/4}$ , we prove that A $A$ determines an asymptotically full proportion of the feasible p $p$ distances. For small sets A $A$ , namely when | A | ⩽ p 4 / 3 $|A|\leqslant p^{4/3}$ over any F $\mathbb {F}$ , we prove that either A $A$ determines ≫ | A | 2 / 3 $\gg |A|^{2/3}$ distances, or A $A$ lies on an isotropic line. For both large and small sets, the results proved are in fact for pinned distances.