Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher-dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances with respect to any flat Riemannian metric. In dimension two they proved a five distance theorem, which is best possible. In this paper, we establish analogous bounds, in all dimensions, for the maximum metric. We also show that in dimensions two and three our bounds are best possible.

中文翻译：

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最大度量的高维间隙定理

最近，第一作者与 Jens Marklof 一起研究了将经典三距离定理推广到高维环面旋转，给出了所有维度上对应于任何平面黎曼度量的距离数的上限。在第二维中，他们证明了一个五距离定理，这是最好的。在本文中，我们在所有维度上为最大度量建立了类似的界限。我们还表明，在维度 2 和维度 3 中，我们的界限是最好的。