On a paucity result in incidence geometry

Abstract The question of the number of collinear tuples in Cartesian set products is a fundamental problem in incidence geometry. We obtain some asymptotic formulas (with power savings in their error terms) for the number of quadruples in the Cartesian product of an arbitrary set A ⊂ ℝ {A\subset{\mathbb{R}}} and for the number of quintuples in A × A {A\times A} for any subset A of the prime field 𝔽 p {\mathbb{F}_{p}} . This is apparently the first such result in incidence geometry. We also obtain some applications of the obtained theorems to incidence problems in 𝔽 p {\mathbb{F}_{p}} .