Completeness in $L^1(R)$ of discrete translates

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra $\mathrm{\Lambda }\subset \mathbb{R}$ for which a generator exists, that is a function $\varphi \in L^1(\mathbb{R})$ such that its $\mathrm{\Lambda }$-translates $\phi(x-\lambda), \lambda\in\Lambda$,span$L^1(\R)$. It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra $\mathrm{\Lambda }\subset \mathbb{R}$ which do not admit a single generator while they admit a pair of generators.