  Let $M$ be a subharmonic function with Riesz measure $\nu_M$ in a domain $D$
in the $n$-dimensional complex Euclidean space $\mathbb C^n$, and let $f$ be a
nonzero function that is holomorphic in $D$, vanishes on a set ${\sf Z}\subset
D$, and satisfies $|f| \leq \exp M$ on $D$. Then restrictions on the growth of
$\nu_M$ near the boundary of $D$ imply certain restrictions on the area/volume
of $\sf Z$. We give a quantitative study of this phenomenon in the subharmonic
framework.
