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

Abstract

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.