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

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a function φ ∈ L1(R) such that its Λ-translates φ(x − λ), λ ∈ Λ, span L1(R). It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra Λ ⊂ R which do not admit a single generator while they admit a pair of generators.