Pólya sequences, Toeplitz kernels and gap theorems

A separated sequence Λ on the real line is called a Pólya sequence if any entire function of zero exponential type bounded on Λ is constant. In this paper we solve the problem by Pólya and Levinson that asks for a description of Pólya sets. We also show that the Pólya–Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and De Branges spaces of entire functions.