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Given a lattice $\Lambda\subseteq\mathbb{Z}^n$ defined by $\Lambda = \{ Mx : x\in\mathbb{Z}^n \}$, let $\Lambda_b$ for $b\in \{0,1\}^n = B$ be the translation of $\Lambda$ by $b$. Call $M$ special whenever $\bigcup_{b\in B}\Lambda_b = \mathbb{Z}^n$. Does there exist a nice characterisation of special matrices?

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    $\begingroup$ These lattices are called "cubiquitous" in low-dimensional topology (arxiv.org/abs/2212.06248). There is some literature on them, but not clear characterisation (as far as I can tell). $\endgroup$
    – Marco Golla
    Commented May 9 at 7:49
  • $\begingroup$ @MarcoGolla Thank you very much for this reference! $\endgroup$
    – mr bean
    Commented May 9 at 10:00
  • $\begingroup$ I had seen the more recent article, but missed the characterisation. Plus I don't know what a Hájos basis is. $\endgroup$
    – Marco Golla
    Commented May 9 at 16:55

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