Nonconstructive proofs of the Beurling–Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions

Two new methods for proving the Beurling–Malliavin theorem on the radius of completeness are given. Development of the first method allows one to obtain new sufficient conditions for a sequence $\Lambda=\{\lambda_n\}\subset\mathbf C$ to be a set of nonuniqueness for a wide class of weighted spaces of entire functions, and development of the second gives conditions for this property to be preserved under small displacements of the points $\lambda_n$.