New bounds on the size of Kakeya sets in finite fields

New bounds on the size of Kakeya sets in finite fields

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Zeev Dvir, IAS
Fine Hall 224

A Kakeya set is a set in $(F_q)^n$ (the $n$ dimensional vector space over a field of $q$ elements) which contains a line in every direction. In this talk I will present a recent result which gives a lower bound of $(q/2)^n$ on the size of such sets. This bound is tight to within a multiplicative factor of two from the known upper bounds. The proof extends the polynomial method used in [Dvir 08, Saraf Sudan 08] and uses polynomials of unbounded degree. If time allows I will also discuss the applications to the explicit construction of mergers that follow from our techniques.Joint work with Kopparty, Saraf and Sudan (arxiv paper: "Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers").