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arXiv · 2011.11225

Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$

Manik Dhar·Zeev Dvir
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Abstract

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

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Manik Dhar, Zeev Dvir. 2020-11-23. Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$. https://doi.org/10.5070/c61055361

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