Metric characteristics of exceptional sets arising in estimates of subharmonic functions

The classes $U_{\mathrm{reg}}$ of subharmonic functions $u(x)$, $x\in\mathbb R^m$, $m\geqslant2$, of finite proximate order are considered, which generalize the class of functions of the form $u(z)=\ln|f(z)|$, where $f(z)$ is an entire function of completely regular growth in the sense of Levin–Pfluger. Estimates are obtained for the exceptional sets $C$ for functions $u(x)\in U_{\mathrm{reg}}$ containing the centers and radii of the balls covering $C$. Coverings of various structures are studied. In particular, the following problem is solved: Under what conditions on a continuous increasing function $h(t)$, $t\geqslant0$, $h(0)=0$, can the set $C$ be covered by balls $B_j(x_j,r_j)=\{x\in\mathbb R^m:|x-x_j|<r_j\}$ such that $\sum_{|x_j|<R}h(r_j/R)=o(1)$ as $R\to\infty$. In an approach proposed by V. S. Azarin these problems reduce to studying the connection between convergence in the topology of the space $\mathscr D'$ of generalized functions and convergence outside the exceptional sets.