Approximate lattices and Meyer sets in nilpotent Lie groups

We show that uniform approximate lattices in nilpotent Lie groups are subsets of model sets. This extends Y.Meyer's theorem about quasi-crystals in Euclidean spaces. We derive from this structure theorem a characterisation of connected, simply connected, nilpotent Lie groups containing approximate lattices as the groups whose Lie algebra have structure constants lying in $\overline{\mathbb{Q}}$.