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

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.