Approximate Lattices and Meyer Sets in Nilpotent Lie Groups

TL;DR

This paper extends Meyer's theorem to nilpotent Lie groups by showing that uniform approximate lattices are subsets of model sets, providing a criterion for their existence based on dense subsets and logarithms.

Contribution

It generalizes Meyer's theorem from Euclidean spaces to nilpotent Lie groups and offers a simple criterion for the existence of approximate lattices in these groups.

Findings

Uniform approximate lattices in nilpotent Lie groups are subsets of model sets.

A criterion for the existence of approximate lattices in nilpotent Lie groups.

Relatively dense subsets and their logarithms are key to the analysis.

Abstract

We show that uniform approximate lattices in nilpotent Lie groups are subsets of model sets. This extends a theorem due to Yves Meyer about quasicrystals 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.