Extensions of Schreiber's theorem on discrete approximate subgroups in $\mathbb{R}^d$

TL;DR

This paper provides an alternative proof of Schreiber's theorem on discrete approximate subgroups in Euclidean space, extends it to the Heisenberg group, and relates these groups to Meyer sets.

Contribution

It offers a new proof of Schreiber's theorem, shows that such subgroups are restrictions of Meyer sets, and extends the theorem to the Heisenberg group.

Findings

Alternative proof of Schreiber's theorem

Discrete approximate subgroups are restrictions of Meyer sets

Extension of the theorem to the Heisenberg group

Abstract

In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $\mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $\mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $\mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.