  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.
