Explicit incidence bounds over general finite fields

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 $q=p^4$.