  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.
