The problem of whether the sumset of a canonical collection covers all of Z is undecidable for a well-behaved family of collections, by equivalence to the universal halting problem for Fractran.
Deciding stability and mortality of piecewise affine dynamical systems
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Additive systems for $\mathbb{Z}$ are undecidable
The problem of whether the sumset of a canonical collection covers all of Z is undecidable for a well-behaved family of collections, by equivalence to the universal halting problem for Fractran.