Lattice coverings and homogeneous covering congruences
J. E. Cremona, P. Koymans
TL;DR
The paper studies lattice coverings of $\ obreakspace\mathbb{Z}^2$ by finitely many cocyclic sublattices of finite index, framing it as a projective analogue of covering systems of congruences. It develops a framework based on cocyclic lattices $L(v;N)$ indexed by the projective line $\,\mathbb{P}(N)$, and defines coverings, irredundancy, minimality, and strong minimality, along with a refinement operation that preserves strong minimality. A key contribution is a Simpson-type lower bound $|\mathcal{C}| \ge 1+G(N)-G(D)$, with $G(p^e)=e(p-1)+1$, which bounds the size of coverings in terms of the index lcm $N$ and divisors $D|N$, and allows finiteness results for fixed sizes. The authors provide complete classifications of minimal lattice coverings up to size 8 (55 for $\le6$, 144 for size 7, and 724 for size 8), with many coverings arising as refinements of the trivial covering and a significant portion not strongly minimal. They also connect the lattice covering problem to the classical covering-congruence problem, offering a projective perspective and density-based reasoning through the notion of weight and primitive vectors.
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.