Stability of shifts, interpolation, and spectrum of atomic measures

We ask which functions $G$ and separated sets $\Gamma$ have the property that the $\Gamma$-shifts of $G$ form an unconditional basis in the $L^p\left(R\right)$-closure of their span for every $p\in [1,\infty ]$. The main result establishes the equivalence of this property to each of the two seemingly unrelated conditions: $\Gamma$ is a set of interpolation for certain Paley–Wiener spaces and the nonexistence of certain measures with given support and spectrum. As a consequence, we answer the question for wide classes of functions $G$ and sets $\Gamma$. In particular, we show the connection between the property and the nonexistence of certain crystalline measures.