  We provide a characterization of discrete sets $\Lambda \subset \mathbb{R}$
that admit a function whose $\Lambda$-translates are complete in the Hardy
space $H^1(\mathbb{R})$. In particular, we show that such a set cannot be
uniformly discrete. We then give a uniformly discrete $\Lambda \subset
\mathbb{R}$ which admits a pair of functions such that their
$\Lambda$-translates are complete in $H^1(\mathbb{R})$.
