On the Pinned Distances Problem in Positive Characteristic

We study the Erd\H os-Falconer distance problem for a set $A\subset $\mathbb{F}$^2$,where$$\mathbb{F}$$isafieldofpositivecharacteristic$p$\mathrm{.}If$$\mathbb{F}$=$\mathbb{F}$_p$andthecardinality$|A|$exceeds$p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $\mathrm{\mid }A\mathrm{\mid }\le p^{4\mathrm{/}3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg \mathrm{\mid }A{\mathrm{\mid }}^{2\mathrm{/}3}$. For both large and small sets, the results proved are in fact for pinned distances.