The Kakeya Set conjecture over Z/NZ for general N

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 ${\mathbb R}^n$ has both Hausdorff dimension and Minkowski dimension equal to $n$. The conjecture is open for $n\ge 3$.

This paper concerns Kakeya sets in the finite setting. In 1999, Wolff conjectured that every Kakeya set in ${\mathbb F}^n$, i.e., a set containing a line in every direction, has size at least $c_n\cdot\lvert {\mathbb F}\rvert^n$. The conjecture was proven by Dvir with $c_n=1/n!$ 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 ${\mathbb Z}/p^k{\mathbb Z}$, and Hickman and Wright over ${\mathbb Z}/N{\mathbb Z}$ 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.