An Improved PointLine Incidence Bound Over Arbitrary Fields
Abstract
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal pointline incidence bound on Cartesian products, using a reduction to a pointplane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sumproducttype problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.06284
 Bibcode:
 2016arXiv160906284S
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 18 pages. To appear in the Bulletin of the London Mathematical Society