Further improvements to incidence and Beck-type bounds over prime finite fields

Abstract

We establish improved finite field Szemeredi-Trotter and Beck type theorems. First we show that if P and L are a set of points and lines respectively in the plane F_p^2, with |P|,|L| \leq N and N<p, then there are at most C_1 N^{3/2-1/662+o(1)} incidences between points in P and lines in L. Here C_1 is some absolute constant greater than 1. This improves on the previously best-known bound of C_1 N^{3/2-1/806+o(1)}. Second we show that if P is a set of points in \mathbb{F}_p^2 with |P|<p then either at least C_2|P|^{1-o(1)} points in P are contained in a single line, or P determines least C_2 |P|^{1+1/109-o(1)} distinct lines. Here C_2 is an absolute constant less than 1. This improves on previous results in two ways. Quantitatively, the exponent of 1+1/109-o(1) is stronger than the previously best-known exponent of 1+1/267. And qualitatively, the result applies to all subsets of F_p^2 satisfying the cardinality condition; the previously best-known result applies only when P is of the form P=A*A for A \subseteq F_p.