On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems

Let $M$ be a subharmonic function on a domain $D\subset \mathbb C^n$ with Riesz measure $\nu_M$, ${\mathsf Z} \subset D$. As was shown in the first of the preceding articles, if there exists a holomorphic function $ f\neq 0 $ on $D$, $f ({\mathsf Z}) = 0$, $|f|\leq \exp M$ on $D$, then there is some scale of integral uniform estimates from above of the distribution of the set $\mathsf Z$ in terms of $\nu_M$. In this article we show that for $n = 1$ this result is “almost invertible”. From such scale estimates of the distribution of points of the sequence ${\mathsf Z}:= \{{\mathsf z} _k \}_{k = 1,2, \dots} \subset D \subset \mathbb C$ by $\nu_M$ it follows that there exists a nonzero holomorphic function $f$ in $D$, $f (\mathsf Z) =0$, $|f| \leq \exp M^{\uparrow}$ on $D$, where the function $ M^{\uparrow} \geq M$ on $D$ is constructed by averaging of $M$ in rapidly convergent disks as we approach the boundary of the domain $D$ with some possible additive logarithmic component associated with the rate of narrowing of these disks.