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

TL;DR

This paper proves a lower bound on the size of Kakeya sets over square-free integers modulo N, extending known results from prime moduli to composite square-free cases, and reduces the problem to rank bounds over finite fields.

Contribution

It establishes the first nontrivial lower bounds for Kakeya sets over composite square-free moduli, generalizing previous prime-only results, and links the problem to incidence matrix rank over finite fields.

Findings

Lower bound of $C_{n, ext{epsilon}} N^{n - ext{epsilon}}$ for Kakeya sets over square-free N

Reduction of the problem to bounding the rank of incidence matrices over finite fields

Extension of Kakeya set bounds from prime to square-free composite moduli

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 $(\mathbb{Z}/p^k\mathbb{Z})^n$.