Work

An improved point-line incidence bound over arbitrary fields

Year: 2017

Type: article

Abstract: We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in F 2 , with m 7 / 8 < n < m 8 / 7 , determine at most O ( m 11 / 15 n 11 / 15 ) incidences (where, if F has positive characteristic p, we assume m − 2 n 13 ≪ p 15 ). 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.

Source: Bulletin of the London Mathematical Society

DOI: 10.1112/blms.12077

Authors: Sophie Stevens, Frank de Zeeuw

Institutions: University of Bristol, École Polytechnique Fédérale de Lausanne

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Awards: Decompositions and Transversals of Geometric Arrangements, Geometric intersection graphs and hypergraphs