The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$

TL;DR

This paper proves the Kakeya set conjecture for the ring of integers modulo N, extending previous results to general N and providing new bounds and constructions for Kakeya sets.

Contribution

It extends the Kakeya set conjecture proof to all $Z/NZ$, combining techniques for prime powers and square-free N, and introduces stronger bounds and near-sharp constructions.

Findings

Proved the Kakeya set conjecture for $Z/NZ$ for general N.

Established stronger lower bounds for $(m,varepsilon)$-Kakeya sets over $Z/p^kZ$.

Provided near-sharp constructions for Kakeya sets over $Z/p^kZ$ and $Z/NZ$.

Abstract

We prove the Kakeya set conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. This entails extending and combining the techniques of Arsovski [Ars21a] for $N=p^k$ and the author and Dvir [DD21] for the case of square-free $N$. We also prove stronger lower bounds for the size of $(m,\epsilon)$-Kakeya sets over $\mathbb{Z}/p^k\mathbb{Z}$ by extending the techniques of [Ars21a] using multiplicities as was done in [SS08, DKSS13]. In addition, we show our bounds are almost sharp by providing a new construction for Kakeya sets over $\mathbb{Z}/p^k\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$.