Properly distributed subsequence on the line

Abstract

In the article we consider first order sequences of complex numbers. We prove that a sequence of nonzero minimal density contains a subsequence of the same density. We also prove that a real sequence of nonzero minimal density contains a properly distributed subsequence. Basing on this fact, we prove a result on representation of an entire function of exponential type with real zeros as a product of two entire functions with the same properties. Moreover, one of these functions has a regular growth. As a corollary, we obtain a result on completeness of exponential systems with real exponents in the space of analytic functions in a bounded convex domain of the complex plane.