Uniqueness distributions for entire functions with uniform constraints on their growth

Let M=Mup-Mlow be the difference of subharmonic functions on the complex plane C. First, we discuse the following general problem: What are the conditions for the distribution of points Z on C, under which there is an entire nonzero function f that vanishes on Z and satisfies the inequality |f|≤eM on C? We formulate some known results for the general problem from one of our papers with co-authors. The next step is to discuss a specific problem of when Mup=b|Im| is the module of the imaginary part with a numerical multiplier b≥0, and Mlow is the Poisson transformation of a positive even function w on the real axis R, increasing on the positive semi-axis R+, and with a finite logarithmic integral. A very significant contribution to this theory is contained in a number of fundamental works by A. V. Abanin, including his known monograph. It is precisely such classes of entire functions that arise after the Fourier--Laplace transform of test functions on compacts. In this direction, the article discusses the limits of applicability of the Beurling--Malliavin theory, and also provides our criterion with co-authors, but only for the zero function w=0. The final main result of the article extends the last criterion to the cases of a nonzero function w≠0.