Exceptional Sets

In this paper, we study sequences of complex numbers of the first order. Multiple terms are allowed for such sequences. We also consider complex sequences with a finite maximum density. We construct special coverings of multiple sets {λ_k, n_k} consisting of circles centered at points λ_k of special radii. In particular, we construct coverings with connected components of a relatively small diameter, as well as coverings that are C₀-sets. These coverings act as exceptional sets for entire functions of exponential type. Outside these sets, we obtain a representation of the logarithm of the modulus of an entire function. Previously, a similar representation was obtained by Levin outside the exceptional set, with respect to which only its existence is asserted. In contrast to this, in this paper we present a simple effective construction of an exceptional set. We construct bases of the invariant subspace of analytic functions in a convex domain. They consist of linear combinations of eigenfunctions and associated functions (exponential monomials) of the differentiation operator divided into relatively small groups.