Open Access Publications from the University of California

## Published Web Location

https://doi.org/10.5070/C61055361
Abstract

A Kakeya set $$S \subset (\mathbb{Z}/N\mathbb{Z})^n$$ is a set containing a line in each direction. We show that, when $$N$$ is any square-free integer, the size of the smallest Kakeya set in $$(\mathbb{Z}/N\mathbb{Z})^n$$ is at least $$C_{n,\epsilon} N^{n - \epsilon}$$ for any $$\epsilon$$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $$N$$. We also show that the case of general $$N$$ can be reduced to lower bounding the $$\mathbb{F}_p$$ rank of the incidence matrix of points and hyperplanes over $$(\mathbb{Z}/p^k\mathbb{Z})^n$$.

Mathematics Subject Classifications: 05B20, 05B25