## 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 $$\mathbb{Z}^d$$.

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 Thursday, March 8, 2018