An improved incidence bound over fields of prime order

TL;DR

This paper establishes a significantly improved upper bound on the number of incidences between points and lines in a finite field of prime order, advancing the understanding of combinatorial geometry in finite fields.

Contribution

The authors present a new incidence bound in (F_p)^2 that improves previous results by an order of magnitude, using novel combinatorial and algebraic techniques.

Findings

Incidence bound of C N^(3/2 - 1/806 + o(1))

Improvement over previous bound by a factor of about 100

Applicable for sets with size up to N < p

Abstract

Let P be a set of points and $L$ a set of lines in (F_p)^2, with |P|,|L|\leq N and N<p. We show that P and L generate no more than C N^(3/2 - 1/806 + o(1)) incidences for some absolute constant C. This improves by an order of magnitude on the previously best-known bound of C N^(3/2 - 1/10678).