On the Distribution of Zero Sets of Holomorphic Functions: III. Converse Theorems

Let M be a subharmonic function in a domain D subset of C-n with Riesz measure nu(M), and let Z subset of D. As was shown in the first of the preceding papers, if there exists a holomorphic function f not equal 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 nu(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,...) subset of D subset of C via nu(M) that there exists a nonzero holomorphic function f in D such that f(Z) = 0 and |f| exp M-up arrow r on D, where the function M-up arrow 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.