Exactness of approximation of a subharmonic function by the logarithm of the modulus of an analytic function in the Chebyshev metric

It is known that a subharmonic function $u(z)$ of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function $f(z)$ at the point $z$ up to $C\log|z|$ outside a very small exceptional set. We prove that if a constant $C$ decreases, then, beginning with the value $C=\rho/4$, the exceptional set enlarges substantially. This improves a result by Yulmukhametov. We also prove similar results for subharmonic functions of infinite order and functions subharmonic in the disk.
The main result of the article is the following.
Theorem 1. Suppose a number $\rho$ is positive, and an entire function $f(z)$ satisfies the condition
$$ ||z|^\rho-\log|f(z)||\le C\log|z|, \qquad z\notin E, $$
where $E\subset\bigcup_j\{z:|z-z_j|<r_j\}$, $r_j<|z_j|^{1-\rho/2-2C+\varepsilon}$, and $\varepsilon>0$. Then
$$ \sum_{R\le|z_j|\le 2R}r_j\ge R^{1+\rho/2-2C-3\varepsilon}, \qquad R>R(\varepsilon). $$