Estimate of the Upper Density of Gabor System

In [1] it was shown that the upper density of a discrete set Λ for which the Gabor system GΛ is complete in the space L2(R) cannot be less than 13π. It is also known from earlier studies that with a regular distribution of indicators, the upper density is not less than 2π. In this paper, we refine the estimate in the absence of the regularity condition for the distribution: the upper density of a discrete set Λ for which the Gabor system GΛ is complete in the space L2(R) cannot be less than 3√4π. Improvement of the estimates is achieved by a more methodical application of symmetrization of this set of indicators of the Gabor system using the known effect of reducing the growth of the modulus of an entire function with a more symmetrical arrangement of its zeros. The possibility of improving the obtained estimate within the proposed method is also discussed using specific examples.