Explicit incidence bounds over general finite fields

TL;DR

This paper establishes new incidence bounds between points and lines over finite fields under specific antifield conditions, advancing understanding of geometric configurations in finite field settings.

Contribution

It introduces a novel incidence bound for antifield point sets over finite fields, extending previous bounds to more general field sizes.

Findings

Proves incidence bound I(P,L) ≤ γ n^{3/2 - 1/12838} for antifield sets.

Provides explicit examples of antifield sets in q=p^2 and q=p^4 cases.

Defines antifield condition as limited interaction with large subfields.

Abstract

Let $\mathbb{F}_{q}$ be a finite field of order $q=p^k$ where $p$ is prime. Let $P$ and $L$ be sets of points and lines respectively in $\mathbb{F}_{q} \times \mathbb{F}_{q}$ with $|P|=|L|=n$. We establish the incidence bound $I(P,L) \leq \gamma n^{3/2 - 1/12838}$, where $\gamma$ is an absolute constant, so long as $P$ satisfies the conditions of being an `antifield'. We define this to mean that the projection of $P$ onto some coordinate axis has no more than half-dimensional interaction with large subfields of $\mathbb{F}_q$. In addition, we give examples of sets satisfying these conditions in the important cases $q=p^2$ and $q=p^4$.