Multi-parameter Szemer\'{e}di-Trotter-type theorems and applications in finite fields

TL;DR

This paper introduces new multi-parameter incidence theorems in finite field vector spaces that outperform existing higher-dimensional results and demonstrates their applications in combinatorial geometry and number theory.

Contribution

It presents novel multi-parameter point-line incidence estimates in finite fields that are more effective than general higher-dimensional results.

Findings

Improved incidence bounds in finite field vector spaces

Applications to combinatorial geometry problems

Applications to number theory problems

Abstract

We prove some novel multi-parameter point-line incidence estimates in vector spaces over finite fields. While these could be seen as special cases of higher-dimensional incidence results, they outperform their more general counterparts in those contexts. We go on to present a number of applications to illustrate their use in combinatorial problems from geometry and number theory.