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I've just spotted this question. It's a well-known open problem in discrete geometry. Unfortunately, I can't remember what it is called.

There is a paper that proves that no union of finitely many lattices can work. I'll try to remember what it is. I remember that it used a result of Mahler.

Added later: I think it was Mahler's selection theorem. Maybe that's enough of a clue for someone to reconstruct the proof (which I seem to remember wasn't hugely hard).

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I've just spotted this question. It's a well-known open problem in discrete geometry. Unfortunately, I can't remember what it is called.

There is a paper that proves that no union of finitely many lattices can work. I'll try to remember what it is. I remember that it used a result of Mahler.