Incidence bounds and applications over finite fields

In this paper we introduce a unified approach to deal with incidence problems between points and varieties over finite fields. More precisely, we prove that the number of incidences $I(\mathcal{P},\mathcal{V})$ between a set $\mathcal{P}$ of points and a set $\mathcal{V}$ of varieties of a certain form satisfies $$\mid I(\mathcal{P},\mathcal{V})-\frac{\mathrm{\mid }\mathcal{P}\mathrm{\mid }\mathrm{\mid }\mathcal{V}\mathrm{\mid }}{q^k}\mid \le q^{dk\mathrm{/}2}\sqrt{\mathrm{\mid }\mathcal{P}\mathrm{\mid }\mathrm{\mid }\mathcal{V}\mathrm{\mid }}\mathrm{.}$$ This result is a generalization of the results of Vinh (2011), Bennett et al. (2014), and Cilleruelo et al. (2015). As applications of our incidence bounds, we obtain results on the pinned value problem and the Beck type theorem for points and spheres. Using the approach introduced, we also obtain a result on the number of distinct distances between points and lines in ${\mathbb{F}}_q^2$, which is the finite field analogous of a recent result of Sharir et al. (2015).