Density results for frames of exponentials

For a separated sequence Λ = {λk}k∈Z of real numbers there is a close link between the lower and upper densities D−(Λ), D(Λ) and the frame properties of the exponentials {ek}k∈Z: in fact, {ek}k∈Z is a frame for its closed linear span in L(−γ, γ) for any γ ∈]0, πD−(Λ)[∪]πD+(Λ),∞[. We consider a classical example presented already by Levinson [10] with D−(Λ) = D(Λ) = 1; in this case, the frame property is guaranteed for all γ ∈]0,∞[\{π}. We prove that the frame property actually breaks down for γ = π. Motivated by this example, it is natural to ask whether the frame property can break down on an interval if D−(Λ) 6= D(Λ). The answer is yes: We present an example of a family Λ with D−(Λ) 6= D(Λ) for which {ek}k∈Z has no frame property in L(−γ, γ) for any γ ∈]πD−(Λ), πD(Λ)[.