Minimal value for the type of an entire function of order $\rho\in(0,1)$, whose zeros lie in an angle and have a prescribed density

Abstract

In the work we find the minimal value that can be taken by the type of an entire function of order $\rho\in(0,1)$ with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than $\pi$. The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A. Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.