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

Research paper by **Timothy G. F. Jones**

Indexed on: **20 Jun '12**Published on: **20 Jun '12**Published in: **Mathematics - Combinatorics**

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#### 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.