Ingham-Beurling type theorems with weakened gap conditions

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CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS (2006, background): “Since then, many generalizations have been given (see, for instance, [10], [104], [8] and [117]).”
STABILIZATION OF SECOND ORDER EVOLUTION EQUATIONS WITH UNBOUNDED FEEDBACK WITH DELAY (2010, unknown): “, M the set of natural numbers km satisfying (see for instance [6]) ⎧⎪⎪⎨ ⎪⎪⎩ λkm − λkm−1 ≥ γ′ 0 λkn − λkn−1 < γ′ 0 for m+ 1 ≤ n ≤ m+ k − 1, λkm+k − λkm+k−1 ≥ γ′ 0.”
An Introduction to the Controllability of Partial Differential Equations (2004, background): “Since then, many generalizations have been given (see, for instance, [6], [58], [4] and [38]).”
A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator. (2005, background): “Note that, due to a misprint, the condition l(I ) > 2' & in the above theorem has been written l(I ) > 2'& in [5].”