Explicit incidence bounds over general finite fields

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$\mathrm{.}Weestablishtheincidencebound$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 $q=p^4$.