On a paucity result in Incidence Geometry

Abstract

We obtain some asymptotic formulae (with power savings in their error terms) for the number of quadruples in the Cartesian product of an arbitrary set $A \subset \mathbf{R}$ and for the number of quintuplets in $A\times A$ for any subset $A$ of the prime field $\mathbf{F}_p$. Also, we obtain some applications of our results to incidence problems in $\mathbf{F}_p$.