Let M be a subharmonic function in a domain D ⊂ ℂⁿ with Riesz measure ν_M, and let Z ⊂ D. As was shown in the first of the preceding papers, if there exists a holomorphic function f ≠ 0 in D such that f(Z) = 0 and |f| ⩽ exp M on D, then one has a scale of integral uniform upper bounds for the distribution of the set Z via ν_M. The present paper shows that for n = 1 this result "almost has a converse." Namely, it follows from such a scale of estimates for the distribution of points of the sequence Z ≔ {z_k}_k=1,2,... ⊂ D ⊂ ℂ via ν_M that there exists a nonzero holomorphic function f in D such that f(Z) = 0 and |f| ⩽ exp M^↑r on D, where the function M^↑r ⩾ M on D is constructed from the averages of M over circles rapidly narrowing when approaching the boundary of D with a possible additive logarithmic term associated with the rate of narrowing of these circles.

中文翻译：

---

关于全纯函数零集的分布：III。逆定理

设M是在一个域中的次谐波功能d ⊂ℂ ^ň与里斯措施ν_中号，并令z⊂ d。如先前论文的第一篇所示，如果D中存在全纯函数f ≠0 ，则f（Z）= 0且| f | ⩽实验值中号上d，则一个具有比例积分均匀的上界为经由所述一组Z的分布ν_中号。本文表明，对于n = 1，此结果“几乎是相反的”。也就是从这样的规模概算序列Z≔{的点的分布的Ž _ķ } _{K = 1,2，...} ⊂ d ⊂ℂ经由ν_中号存在一个非零全纯函数˚F在d使得˚F（Z）= 0和 f | ⩽EXP中号^{↑ - [R}上d，其中函数中号^{↑ - [R} ⩾中号上d是从平均值构造中号超过圈接近的边界时迅速缩小d 以及与这些圆变窄的速率相关的可能的对数项。