A remark on dimensionality reduction in discrete subgroups

In this short note, we prove a version of the Johnson-Lindenstrauss flattening Lemma for point sets taking values in discrete subgroups. More precisely, given $d,{\lambda }_0,N_0\in \mathbb{N}$ and $\epsilon \in (0,\frac{1}{2})$ suitably chosen, we show there exists a natural number $k=k(d,\epsilon )=O(\frac{1}{{\epsilon }^2}\log d)$, such that for every sufficiently large scaling factor $\lambda \in \mathbb{N}$ and any point set $\mathcal{D}\subset \frac{\lambda }{{\lambda }_0}{\mathbb{Z}}^d\cap B(0,\lambda N_0)$ with cardinality $d$, there exists an embedding $F:\mathcal{D}\to \frac{1}{{\lambda }_0}{\mathbb{Z}}^k$, with distortion at most $(1+\epsilon +\frac{\epsilon }{\lambda {\lambda }_0})$.