Completeness in L¹(R) of discrete translates

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra $\Lambda\subset\R$ for which a generator exists, that is a function $\phi\in L^1(\R)$ such that its $\Lambda$-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 $\Lambda\subset\R$ which do not admit a single generator while they admit a pair of generators.