齐次动力学下的高维Steinhaus和Slater问题
Higher dimensional Steinhaus and Slater problems via homogeneous dynamics
摘要：
三间隙定理，也被称为斯坦豪斯猜想或三距离定理，指出$α,2α,\ldots, Nα$的分数部分的间隙最多取三个不同的值。激发的问题Erdés，Geelen和辛普森，我们探讨了一个更高的维度的变体，它要求的数量之间的差距的分数部分的线性型。利用格空间上对角作用的遍历性质，我们证明了对于几乎所有的参数值，高维问题中的不同间隙数是无界的。我们的结果特别是提高早期的工作由Boshernitzan，戴森和Bleher等人。我们进一步讨论了一个密切的联系与Littlewood猜想在乘法丢番图逼近。最后，我们还演示了如何调整我们的方法以获得类似的结果，即平移到更高维度环面上缩小区域的返回时间之间的间隙。
英文摘要：
The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of $α,2α,\ldots, Nα$ take at most three distinct values. Motivated by a question of Erdős, Geelen and Simpson, we explore a higher-dimensional variant, which asks for the number of gaps between the fractional parts of a linear form. Using the ergodic properties of the diagonal action on the space of lattices, we prove that for almost all parameter values the number of distinct gaps in the higher dimensional problem is unbounded. Our results in particular improve earlier work by Boshernitzan, Dyson and Bleher et al. We furthermore discuss a close link with the Littlewood conjecture in multiplicative Diophantine approximation. Finally, we also demonstrate how our methods can be adapted to obtain similar results for gaps between return times of translations to shrinking regions on higher dimensional tori.
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。
