  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.

Number Theory;Dynamical Systems;Metric Geometry