On infinite discrete approximate subgroups in \(\mathbb{R}^d\)

Alexander Fish

Abstract

In this paper we show that any discrete, infinite approximate subgroup \(\Lambda \subset \mathbb{R}^d\) is relatively dense around some linear subspace \(L \subset \mathbb{R}^d\), i.e., there exists \(R > 0\) such that for every ball \(B_R(x)\) with center at \(x \in L\) we have \(\Lambda \cap B_R(x) \neq \emptyset \), and \(\Lambda \subset \cup_{x \in L} B_R(x)\). As an application of our main theorem, we provide a complete classification of infinite approximate subgroups in \(\mathbb{Z}^d\).