Stability of shifts, interpolation, and spectrum of atomic measures

We ask which functions and separated sets have the property that the ‐shifts of form an unconditional basis in the ‐closure of their span for every . The main result establishes the equivalence of this property to each of the two seemingly unrelated conditions: 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 and sets . In particular, we show the connection between the property and the nonexistence of certain crystalline measures.