2016/09/30 by Sophie Stevens, Frank de Zeeuw · 1 citation
Mathematics · #math.CO
paper · pdf · doi:10.1112/blms.12077
18 pages. To appear in the Bulletin of the London Mathematical Society
arxiv created 2017/07/18 · arxiv updated 2017/08/16
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field \mathbbF, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in \mathbbF2, with m7/8<n<m8/7, determine at most O(m11/15n11/15) incidences (where, if \mathbbF has positive characteristic p, we assume m-2n13≪ p15). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane 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 sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.