Higher dimensional gap theorems for the maximum metric

TL;DR

This paper extends classical gap theorems to higher dimensions using the maximum metric, establishing bounds on the number of distances for toral rotations, with optimal results in dimensions two and three.

Contribution

It provides new upper bounds for the number of distances in higher-dimensional toral rotations under the maximum metric, generalizing previous results and proving optimality in low dimensions.

Findings

Established bounds for the number of distances in all dimensions.

Proved the bounds are optimal in dimensions two and three.

Extended classical three distance theorem to the maximum metric.

Abstract

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.