## 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 language | English |
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Pages (from-to) | 842-858 |

Number of pages | 17 |

Journal | Bulletin of the London Mathematical Society |

Volume | 49 |

Issue number | 5 |

Early online date | 3 Aug 2017 |

DOIs | |

Publication status | Published - Oct 2017 |