Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$

A Kakeya set $S\subset (\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^n$ 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 $(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^n$ is at least $C_{n,ϵ}{N}^{n-ϵ}$ for any $ϵ$ – 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 $(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z}{)}^n$.