Maximal and $(m,\epsilon)$-Kakeya bounds over $\mathbb{Z}/N\mathbb{Z}$ for general $N$

TL;DR

This paper proves the Maximal Kakeya conjecture over $\

Contribution

It establishes Maximal Kakeya bounds over $\\mathbb{Z}/N\\mathbb{Z}$ for general $N$, extending previous results and providing new proofs and bounds.

Findings

Proved Maximal Kakeya estimates for $\\mathbb{Z}/N\\mathbb{Z}$

Derived lower bounds for $(m,\epsilon)$-Kakeya sets

Provided a new proof for finite field Kakeya bounds

Abstract

We derive Maximal Kakeya estimates for functions over $\mathbb{Z}/N\mathbb{Z}$ proving the Maximal Kakeya conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. The proof involves using polynomial method and linear algebra techniques from [Dha21, Ars21a, DD21] and generalizing a probabilistic method argument from [DD22]. As another application we give lower bounds for the size of $(m,\epsilon)$-Kakeya sets over $\mathbb{Z}/N\mathbb{Z}$. Using these ideas we also give a new, simpler, and direct proof for Maximal Kakeya bounds over finite fields (which were first proven in [EOT10]) with almost sharp constants.