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Publication - Miss Sophie Stevens

    An improved point‐line incidence bound over arbitrary fields

    Citation

    Stevens, S & Zeeuw, Fd, 2017, ‘An improved point‐line incidence bound over arbitrary fields’. Bulletin of the London Mathematical Society, vol 49., pp. 842-858

    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.

    Full details in the University publications repository