On the accuracy of asymptotic approximation of subharmonic functions by the logarithm of the modulus of an entire function

We study the degree of possible accuracy of the asymptotic approximation of subharmonic functions by the logarithm of the modulus of an entire function. It is proved that if a subharmonic function $u$ is twice differentiable and satisfies the condition
$$ m\le|z|\Delta u(z)\le M,\qquad|z|>0, $$
where $M,m>0$, then approximation with the accuracy $q\ln|z|+O(1)$ with the constant $q\in(0,\frac14)$ is possible only outside sets of non-$C_0$-set. On the other hand, it is shown that approximation with the accuracy to $q\ln|z|+O(1)$ with the constant $q\ge\frac14$ is possible outside sets, that can be covered by circles $B(z_k,r_k)$ so that
$$ \sum_{|z_k|\le R}r_k=O(R^{\frac34-q}) $$
when $q\in\bigl[\frac14,\frac34\bigr]$ and
$$ \sum_{|z_k|\ge R}r_k=O(R^{\frac34-q}) $$
when $q>\frac34$. In particular, these sets are $C_0$-sets when $q>\frac14$. In the second case, the approximating function is the same for all $q\ge\frac14$, and this function is only a small modification of sine type functions, constructed by Yu. Lubarsky and M. Sodin.