Approximate lattices and Meyer sets in nilpotent Lie groups

Abstract

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 $\overline{\mathbb{Q}}$.