Geodesic
Exceptional sets of entire functions of completely regular growth
Matematičeskie zametki, Tome 116 (2024) no. 4, pp. 510-530

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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.
Keywords: refined order, entire function, regular growth, exceptional set. 
A. S. Krivosheev; O. A. Krivosheeva. Exceptional sets of entire functions of completely regular growth. Matematičeskie zametki, Tome 116 (2024) no. 4, pp. 510-530. http://geodesic.mathdoc.fr/item/MZM_2024_116_4_a2/
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