Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions

Abstract

The idea to write this note arose during the discussion that followed the report of the first author at the International Scientific Conference "Ufa Autumn Mathematical School-2022". We propose three general methods for constructing uniqueness sets in classes of entire functions with growth restrictions. In all three cases, the sequence of zeros of an entire function with special properties is chosen as such set. The first method is related to Sylvester famous problem on the smallest circle containing a given set of points on a plane, and theorems of convex geometry. The second method initially relies on Helly theorem on the intersection of convex sets and its application to the possibility of covering one set by shifting another. The third method is based on the classical Jensen formula, which allows one to estimate the type of an entire function via the averaged upper density of the sequence of its zeros. We present only basic results now. The development of our approaches is expected to be presented in subsequent works.