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

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 $\mathbb{Z}/N\mathbb{Z}$.

To be published in Advances in Combinatorics 2024:2, 26pp

References in corpus (3)

• Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$
• Simple proofs for Furstenberg sets over finite fields
• Maximal and $(m,\epsilon )$-Kakeya bounds over $\mathbb{Z}/N\mathbb{Z}$ for general $N$