High Dimensional Variants of the Finite Field Kakeya Problem

The finite field Kakeya problem asks about the size of the smallest set in (F_q)^n containing a line in every direction.  Raised by Wolff in 1999 as a ‘toy’ version of the Euclidean Kakeya conjecture, this problem is now completely resolved using the polynomial method. In this talk I will describe recent progress on its higher dimensional variant in which lines are replaced with k-dimensional flats. It turns out that, unlike in the one dimensional case, when k >=2,  one can prove that there are no 'interesting’ constructions (with size smaller than trivial) even if one asks for sets that only have large intersection with a flat in every  direction.  This theorem turns out to have surprising applications in questions involving Lattice coverings and linear hash functions.

Based on joint works with Ben Lund and Manik Dhar.