The Kronecker-Weyl equidistribution theorem and geodesics in 3-manifolds

TL;DR

This paper establishes a connection between algebraic number theory and the uniform distribution of geodesics in cube-tiled 3-manifolds, providing explicit directions for such distributions.

Contribution

It introduces explicit directions based on cubic algebraic numbers that ensure uniform distribution of geodesics in cube-tiled 3-manifolds, linking number theory and geometric topology.

Findings

Infinitely many explicit directions lead to uniformly distributed geodesics.

Directions are related to cubic algebraic numbers.

Results apply to any rectangular polyhedron 3-manifold tiled with cubes.

Abstract

Given any rectangular polyhedron 3-manifold $P$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in $P$.