An Interpretation of the Beurling-Malliavin Theorem on the Radius of Completeness

Gap Theorem for Separated Sequences without Pain

“…There exist at least five definitions of the upper Beurling-Malliavin density (see paper [4] which is devoted to equivalence of different definitions). We start with the most wellknown: Definition 1.…”

On the upper and lower exponential density functions

“…Later the same quantity b(Λ) was studied by other authors, for instance [8], [9], and used in several papers; we cite [7], [10] and [15]; for an extensive study and generalizations see [13], [12], [1] (see [14] for a complete list of references); in particular the author of [15] discovers an alternative equivalent definition, which will be stated and used later (see Definition 2.1).…”

Minimum of modulus of the sum of Dirichlet series converging in a half-plane

“…The asymptotic estimate of quantity ( ) for the sum of entire Dirichlet series (1) in terms of maximum | | on the vertical segment = { = + : | − | } of a certain length was obtained in [1]. At that, the length | | of segment should not be less than a certain characteristics close to similar characteristics allowing one to determine the completeness radius for the system of exponents {︀ }︀ in space [ , ] (or 2 [ , ]) (on this subject see [1,4,5,6]). It is natural to expect that while estimating ( ) by the minimum of modulus, in a general situation the length of segment can not be too large.…”