Almost integer translates. Do nice generators exist?

It has been shown earlier by the first author that for any nonzero perturbation of the integers λ_n = n + 0(1), λ_n ≠ n, there is a generator, that is a function ψ ∈ L²(R) such that the system of translates {ψ(x - λ_n)} is complete in L²(R). We ask if ψ can be chosen with fast decay. We prove that in general it cannot. On the other hand, if the perturbations are 'quasianalytically small,' than it can, and this decay restriction is sharp. A certain class of complex measures which we call 'shrinkable' is introduced, and it is shown that the zeros sets of such measures do dot admit generators with fast decay.