Uniformity in cube-covering systems

TL;DR

This paper extends equidistribution theorems to higher-dimensional flat manifolds, addressing challenges in geodesic flow analysis beyond two dimensions with new methods.

Contribution

It develops higher-dimensional analogs of the Kronecker-Weyl theorem, advancing understanding of geodesic flow in complex flat manifolds.

Findings

Higher-dimensional equidistribution results established

New approach for non-integrable flat systems in dimensions >2

Addresses challenges in geodesic flow analysis

Abstract

We establish various analogs of the Kronecker-Weyl equidistribution theorem that can be considered higher-dimensional versions of results established in our earlier investigation of the discrete 2-circle problem studied in 1969 by Veech. Whereas the Veech problem can be viewed as one of geodesic flow on a 2-dimensional flat surface, here we study geodesic flow in higher-dimensional flat manifolds. This is more challenging, as the overwhelming majority of the available proof techniques for non-integrable flat systems are based on arguments in dimension 2. For higher dimensions, we need a new approach.