Proof of the Kakeya set conjecture over rings of integers modulo square-free N
Proof of the Kakeya set conjecture over rings of integers modulo square-free N
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DOI：
10.5070/c61055361
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发表时间：
2020-11
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影响因子：
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通讯作者：
Manik Dhar;Zeev Dvir
中科院分区：
文献类型：
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作者：
Manik Dhar;Zeev Dvir
一个Kakeya集合$S \subset（\mathbb{Z}/N\mathbb{Z}）^n$是一个在每个方向上都包含一条直线的集合。我们证明了，当$N$是任意无平方因子的整数时，$（\mathbb{Z}/N\mathbb{Z}）^n$中的最小Kakeya集的大小至少是$C_{n，\displaystyle\mathbb{Z} N^{n - \times}$，对任意的$\times $ --解决了Hickman和Wright猜想的一个特例.以前，这样的界限只知道素数$N$的情况。我们还证明了一般$N$的情况下，可以减少到下界的点和超平面的关联矩阵的$\mathbb{F}_p$秩$（\mathbb{Z}/p^k\mathbb{Z}）^n$。
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 $(\mathbb{Z}/p^k\mathbb{Z})^n$.