球面上格点的局部统计
Local statistics of lattice points on the sphere
摘要：
一个著名的结果勒让德和高斯确定哪些整数可以表示为三个平方和，并为那些它是典型的情况下，有很多方式这样做。这些不同的陈述给集合点的单位领域，和一个基本的结果，阐明了林尼克，是在一个简单的条件下，这些成为均匀分布的领域。在这篇文章中，我们回顾了我们最近的一些工作，这些工作探索了均匀分布之外发生的事情，为较小尺度上的随机性提供了证据。我们处理静电能量，局部统计，如点对统计（里普利函数），最近邻统计，最小间距和覆盖半径。我们简单地讨论一下其他维度的情况，这是非常不同的。在附录中，我们计算随机点的相应量
英文摘要：
A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points
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。
