A Five Distance Theorem for Kronecker Sequences
A Five Distance Theorem for Kronecker Sequences
复制标题
克罗内克序列的五距离定理
DOI：
10.1093/imrn/rnab205
复制
发表时间：
2021
影响因子：
1
通讯作者：
Marklof, Jens
中科院分区：
文献类型：
--
作者：
Haynes, Alan;Marklof, Jens
三距离定理（也称为三间隙定理或Steinhaus问题）指出，对于任何给定的实数和整数，Kronecker序列中相邻元素之间的距离最多有三个值。在本文中，我们考虑了三距离定理在高维整数格上的Kronecker序列模的自然推广。我们证明了在2D中，对于所有and的选择，最近邻之间的距离最多有5个值。此外，对于几乎每一个，确实有5个不同的距离对无穷多个出现，因此5是最好的可能的一般上界。在更高的维度中，我们有类似的显式上界，但不那么精确。例如，在3D中，我们的边界是13，尽管我们推测真实值是9。我们进一步研究了在一个受限的方向锥中，从一点到它最近的邻居的可能距离的个数。这可以看作是一维中间隙长度的推广。对于大的锥角，我们使用几何参数来产生直接类似于三距离定理的显式边界。对于小锥角，我们利用非模格空间中齐次流的遍历理论，证明了不同长度的数目(1)对于几乎所有的都是无界的，(2)对于满足某些丢芬图条件的都是有界的。
The three-distance theorem (also known as the three-gap theorem or Steinhaus problem) states that, for any given real numberand integer, there are at most three values for the distances between consecutive elements of the Kronecker sequencemod 1. In this paper, we consider a natural generalization of the three-distance theorem to the higher-dimensional Kronecker sequencemodulo an integer lattice. We prove that in 2D, there are at most five values that can arise as a distance between nearest neighbors, for all choices ofand. Furthermore, for almost every, five distinct distances indeed appear for infinitely manyand hence five is the best possible general upper bound. In higher dimensions, we have similar explicit, but less precise, upper bounds. For instance, in 3D, our bound is 13, though we conjecture the truth to be 9. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalization of the gap length in 1D. For large cone angles, we use geometric arguments to produce explicit bounds directly analogous to the three-distance theorem. For small cone angles, we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (1) unbounded for almost alland (2) bounded forthat satisfy certain Diophantine conditions.