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Uniformity in cube-covering systems

J. Beck, W. W. L. Chen

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.

Uniformity in cube-covering systems

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.

Paper Structure

This paper contains 6 sections, 19 theorems, 350 equations.

Table of Contents

  1. Analog of the Veech $2$-circle problem
  2. Preparation for the proof of Theorem \ref{['thm4']}
  3. Proof of Theorem \ref{['thm4']}
  4. Proof of Lemmas \ref{['lem32']}, \ref{['lem35']} and \ref{['lem36']}
  5. Establishing Lemma \ref{['lem34']}
  6. Piecewise smooth barriers

Key Result

Theorem \oldthetheorem

Let $n\geqslant2$ be an integer, and let $\mathcal{M}$ be any $n$-cube $3$-manifold with barriers, where the $yz$-parallel square faces have identical $2$-coloring such that each of the red and green parts is the union of finitely many polygons, and where the green part has positive area. Then for a

Theorems & Definitions (34)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 24 more