球面 $\mathbb{S}^2$ 上具有小球冠差异的点集
Point sets on the sphere $\mathbb{S}^2$ with small spherical cap discrepancy
AI总结：
本文研究二维单位球面上均匀分布点集的几何差异，证明随机点集、球面数字网和球面斐波那契格点的球冠差异收敛阶为 $N^{-1/2}$，并通过保面积Lambert映射分析预像的等高线。
摘要：
本文研究二维单位球面上显式构造的均匀分布点集的几何差异。我们证明了随机点集、球面数字网和球面斐波那契格点的球冠差异收敛阶为 $N^{-1/2}$。因此，这些点集对于数值积分和其他计算模拟非常有用。证明使用了保面积的Lambert映射。文中给出了该映射下球冠预像的等高线和集合的详细分析。
英文摘要：
In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given.
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。
