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arXiv · 2110.14889

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

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

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BibTeXRIS

Manik Dhar. 2021-10-28. The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$. https://doi.org/10.19086/aic.2024.2

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