In this paper we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J. Brauchart, P. Grabner, W. Kusner and J. Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls, or balls of threshold order on the flat tori are uniformly distributed. Moreover, it is also shown that QMC-designs sequences for Sobolev classes, probabilistic point sets (obtained from jittered samplings), and some determinantal point process are hyperuniform.

中文翻译：

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Flat Tori 上的超均匀点集：确定性和概率方面

在本文中，我们研究了平面圆环上的超均匀性。J. Brauchart、P. Grabner、W. Kusner 和 J. Ziefle 已经研究了单位球面上的超均匀点集。结果表明，对于平坦环面上的大球、小球或阈值顺序球而言，超均匀的点集是均匀分布的。此外，还表明 QMC 设计的 Sobolev 类序列、概率点集（从抖动采样中获得）和一些行列式点过程是超均匀的。