On the pinned distances problem in positive characteristic

We study the Erdos-Falconer distance problem for a set A subset of F2$A\subset \mathbb {F}<^>2$, where F$\mathbb {F}$ is a field of positive characteristic p$p$. If F=Fp$\mathbb {F}=\mathbb {F}_p$ and the cardinality |A|$|A|$ exceeds p5/4$p<^>{5/4}$, we prove that A$A$ determines an asymptotically full proportion of the feasible p$p$ distances. For small sets A$A$, namely when |A|<= p4/3$|A|\leqslant p<^>{4/3}$ 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.

Automated summary

The Erdos-Falconer distance problem for a set A in F2 is studied, where F is a field with positive characteristic p. If F=Fp and the cardinality of A exceeds p5/4, it is proven that A determines an asymptotically full proportion of the feasible p distances. For small sets A, namely when |A|<= p4/3 over any F, it is proven that either A determines >>|A|2/3 distances, or A lies on an isotropic line. The results proved are for pinned distances in both large and small sets.