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 Yves Meyer's theorem about quasi-crystals in Euclidean spaces. To do so we study relatively dense subsets of simply connected nilpotent Lie groups and their logarithms. We then deduce a simple criterion for the existence of an approximate lattice in a given nilpotent Lie group.