On a paucity result in Incidence Geometry

TL;DR

This paper derives asymptotic formulas with power savings for counting specific quadruples and quintuplets in sets over real numbers and prime fields, with applications to incidence problems in finite fields.

Contribution

It introduces new asymptotic formulas with error bounds for counting configurations in sets over real numbers and prime fields, advancing incidence geometry understanding.

Findings

Asymptotic formulas with power savings for quadruples in real sets.

Asymptotic formulas for quintuplets in prime fields.

Applications to incidence problems in finite fields.

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 $\mathbf{F}_p$.