Szemerédi-Trotter type results in arbitrary finite fields

Abstract

Let $q$ be a power of a prime and ${\mathbb{F}}_q$ the finite field consisting of $q$ elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in ${\mathbb{F}}_q^2$. We also use our results on point-line incidences to give new sum-product type estimates concerning sums of reciprocals.

25 Pages