Completeness of Discrete Translates in $H^1(\mathbb{R})$

We provide a characterization of discrete sets $\mathrm{\Lambda }\subset \mathbb{R}$ that admit a function whose $\mathrm{\Lambda }$-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}$$whichadmitsapairoffunctionssuchthattheir$\Lambda$-translatesarecompletein$H^1($\mathbb{R}$)$.