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arXiv · 2601.03212

Lattice coverings and homogeneous covering congruences

J. E. Cremona·P. Koymans
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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.

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J. E. Cremona, P. Koymans. 2026-01-06. Lattice coverings and homogeneous covering congruences. https://arxiv.org/abs/2601.03212

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