A Kakeya set is a set of points of an Euclidean space that contains a unit line segment in every direction. In 1919, Besicovitch constructed a Kakeya set of measure zero for every dimension, and in addition, he constructed sets in the plane with arbitrarily small measure such that a unit segment can rotate full 360 degrees within the set. While Kakeya sets can have measure zero, the famous Kakeya Conjecture asserts that every Kakeya set in Rn has both Hausdorff dimension and Minkowski dimension equal to n. The conjecture is open for n≥3.
This paper concerns Kakeya sets in the finite setting. In 1999, Wolff conjectured that every Kakeya set in Fn, i.e., a set containing a line in every direction, has size at least cn⋅|F|n. The conjecture was proven by Dvir with cn=1/n! in 2008 (an exposition of the proof can be found in this blog post on Terence Tao’s blog). Subsequently, Ellenberg, Oberlin and Tao proposed studying Kakeya sets over the rings Z/pkZ, and Hickman and Wright over Z/NZ for an arbitrary N. The paper resolves a conjecture of Hickman and Wright by giving an exponential lower bound on the size of a Kakeya set in this most general setting.