Exceptional sets of entire functions of completely regular growth

In this paper, we study sequences of complex numbers of refined order. Multiple terms are allowed in such sequences. We consider complex sequences with finite maximal density for a given refined order. We construct special coverings of multiple sets $\{\lambda_k,n_k\}$ consisting of circles of special radii centered at points $\lambda_k$. In particular, we construct coverings whose connected components have a relatively small diameter, as well as coverings that are $C_0$-sets. These coverings act as exceptional sets for entire functions of finite refined order and completely regular growth. Outside these sets, we obtain a representation of the logarithm of the modulus of an entire function. Earlier, a similar representation was obtained by B. Ya. 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 constructive construction of the exceptional set.