Combinatorial Theory

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