We study the dispersion of point sets in the unit square; i.e. the size of the largest axes-parallel box amidst such point sets. It is known that ${lim\; inf}_{N\to \infty }N\mathrm{disp}(N,2)\in \left[\frac{5}{4},2\right],$ where $\mathrm{disp}(N,2)$ is the minimal possible dispersion for an $N$-element point set in the unit square. The upper bound 2 is obtained by an explicit point construction—the well-known Fibonacci lattice. In this paper we find a modification of this point set such that its dispersion is significantly lower than the dispersion of the Fibonacci lattice. Our main result will imply that ${lim\; inf}_{N\to \infty }N\mathrm{disp}(N,2)\le {\phi }^3∕\sqrt{5}=1.894427...$

中文翻译：

---

改进的斐波那契晶格的改进色散界

我们研究点集在单位平方中的离散度；即，在这些点集中的最大平行轴框的大小。众所周知${lim\; inf}_{ñ\to \infty }ñ\mathrm{显示}（ñ，2）\in \left[\frac{5}{4}，2\right]，$ 哪里 $\mathrm{显示}（ñ，2）$ 是对于 $ñ$-在单位正方形中设置的元素点。上限2是通过显式点构造-众所周知的斐波那契晶格获得的。在本文中，我们发现了对该点集的修改，以使其散布明显低于斐波那契晶格的散布。我们的主要结果将暗示${lim\; inf}_{ñ\to \infty }ñ\mathrm{显示}（ñ，2）\le {\phi }^3∕\sqrt{5}=\mathrm{1个}。894427。。。$