改进的Fibonacci格的色散界
Improved dispersion bounds for modified Fibonacci lattices
摘要：
我们研究了单位正方形中点集的分散性，即在这些点集中最大的平行轴盒的大小。已知$\liminf_{N\to\infty} N\mathrm{disp}(N,2)\in \left[\frac54,2\right],$，其中$\mathrm{disp}(N,2)$是单位正方形中$N$元素点集的最小可能色散。上界2是由一个明确的点结构-著名的斐波那契格。在本文中，我们发现这个点集的修改，使其色散显着低于色散的斐波那契晶格。我们的主要结果表明，$\liminf_{N\to\infty} N\mathrm{disp}(N,2)\leq φ^3/\sqrt{5}=1.894427...$
英文摘要：
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 $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\in \left[\frac54,2\right],$ 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 $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\leq φ^3/\sqrt{5}=1.894427...$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
