An improved point‐line incidence bound over arbitrary fields

Sophie Stevens, Frank de Zeeuw

Research output: Contribution to journalArticle (Academic Journal)peer-review

28 Citations (Scopus)
240 Downloads (Pure)

Abstract

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field 픽, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in 픽2, with m7/8<n<m8/7, determine at most O(m11/15n11/15) incidences (where, if 픽 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.

Original languageEnglish
Pages (from-to)842-858
Number of pages17
JournalBulletin of the London Mathematical Society
Volume49
Issue number5
Early online date3 Aug 2017
DOIs
Publication statusPublished - Oct 2017

Bibliographical note

18 pages. To appear in the Bulletin of the London Mathematical Society

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