Lattice coverings and homogeneous covering congruences

TL;DR

This paper studies how to cover the integer lattice d7d72 with a finite set of sublattices, exploring minimal coverings, their construction, and classification for small numbers of sublattices, connecting to classical covering systems.

Contribution

It introduces a projective perspective on lattice coverings, provides a construction method for minimal coverings, and classifies all minimal coverings with up to 8 sublattices.

Findings

Constructed many minimal coverings of d7d72 with sublattices.

Determined all minimal coverings with at most 8 sublattices.

Connected lattice covering problems to classical covering systems.

Abstract

We consider the problem of covering $\mathbb{Z}^2$ with a finite number of sublattices of finite index, satisfying a simple minimality or non-degeneracy condition. We show how this problem may be viewed as a projective (or homogeneous) version of the well-known problem of covering systems of congruences. We give a construction of minimal coverings which produces many, but not all, minimal coverings, and determine all minimal coverings with at most $8$ sublattices.