The restricted variable Kakeya problem (Pete Clark, University of Georgia)

For a finite field F_q, a subset of F_q^N is a Kakeya set if it contains a line in every direction (i.e., a coset of every one-dimensional linear subspace).  The finite field Kakeya problem is to determine the minimal size K(N,q) of a Kakeya set in F_q^N.  This problem was posed by Wolff in 1999 as an analogue to the Kakeya problem in Euclidean N-space, which was (and still is) one of the major open problems in harmonic analysis.  It caused quite a stir in 2008 when Zeev Dvir showed that for each fixed N, as q -> oo, K(N,q) is bounded below by a constant times q^N: the Euclidean analogue of this result is not only proved but known to be false.

But what about the constant?  In 2009 Dvir-Kopparty-Saraf-Sudan gave a lower bound on K(N,q) that was within a factor of 2 of an upper bound due to Dvir-Thas.  (I will briefly mention recent work of Bukh-Chao giving a decisive further improvement, but that is not the focus of the talk.) The key to this improved lower bound is a multiplicity enhancement of a 1922 result of Ore. In this talk I want to give my own exposition of this work together with a mild generalization: if X is a subset of F_q^N \ {0}, then an X-Kakeya set is a subset that contains a translate of the line generated by x for all x in X.  Putting K_X(N,q) to be the minimal size of an X-Kakeya set in F_q^N, I will give a lower bound on K_X(N,q) that recovers the DKSS bound when X = F_q^N \ {0}.  This is similar in spirit to  “statistical Kakeya” results of Dvir and DKSS but not overlapping much; in fact, I will give a statistical generalization of my result as well.