Erdős conjectured that a set of $n$ points in the Euclidean plane has at least $c n/\sqrt{\log n}$ distinct distances for some universal constant $c>0$. Guth and Katz nearly resolved this question, but many related problems remain wide open. I will discuss recent results on a variant of this problem for points in a plane over a finite field.
This is joint work with Giorgis Petridis.