Improved lower bound on the size of Kakeya sets over finite fields

Abstract

In a recent breakthrough, Dvir showed that every Kakeya set in F must be of cardinality at least cn|F| where cn ≈ 1/n!. We improve this lower bound to β|F| for a constant β > 0. This pins down the growth of the leading constant to the right form as a function of n. Let F be a finite field of q elements. Definition 1 (Kakeya Set) A set K ⊆ F is said to be a Kakeya set in F, if for every b ∈ F, there exists a point a ∈ F such that for every t ∈ F, the point a+ t · b ∈ K. In other words, K contains a “line” in every “direction”. The question of establishing lower bounds on the size of Kakeya sets was posed in Wolff [7]. Till recently, the best known lower bound on the size of Kakeya sets was of the form q for some α < 1. In a recent breakthrough Dvir [1] showed that every Kakeya set must have cardinality at least cnq for cn = (n!)−1. (Dvir’s original bound achieved a weaker lower bound of cn · qn−1, but [1] includes the stronger bound of cn · q, with the improvements being attributed to Alon and Tao.)