论文标题
Kakeya设定的猜想的证明，整数Modulo Square $ n $
Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$
论文作者
论文摘要
kakeya集$ s \ subset（\ mathbb {z}/n \ mathbb {z}）^n $是一个在每个方向上包含一条线的集合。我们表明，当$ n $是任何无平方的整数时，$（\ mathbb {z}/n \ mathbb {z}）^n $设置为最小的Kakeya的大小至少是$ c_ {n，ε} n^{n^{n^{n^{n -am} $ for Any $ε$ - 解决的cas an a nick a nickman and nickman and tright。以前，此类界限仅因Prime $ n $而闻名。我们还表明，可以将一般$ n $的情况降低为降低$ \ mathbb {f} _p $点的发射机矩阵和超过$（\ mathbb {z}/p^k \ mathbb {z z}）^n $的发射机矩阵和超越平面的等级。
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 $C_{n,ε} N^{n - ε}$ for any $ε$ -- 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 $(\mathbb{Z}/p^k\mathbb{Z})^n$.
