Geodesic
The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$
Advances in Combinatronics (2024)

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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 $\mathbb{Z}/N\mathbb{Z}$.
Publié le : 
@article{ADVC_2024_a5,
     author = {Manik Dhar},
     title = {The {Kakeya} {Set} {Conjecture} for $\mathbb{Z}/N\mathbb{Z}$ for general $N$},
     journal = {Advances in Combinatronics},
     publisher = {mathdoc},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADVC_2024_a5/}
}
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Manik Dhar. The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$. Advances in Combinatronics (2024). http://geodesic.mathdoc.fr/item/ADVC_2024_a5/