## A Blaschke-type product and random zero sets for Bergman spaces.(English)Zbl 0764.30029

This paper provides significant new information about zero sets of analytic functions which are in the Bergman spaces $$\mathbb{A}^ p$$ of functions in the unit disc. For each $$\lambda$$, $$|\lambda|<1$$ a Blaschke-type factor $$f_ \lambda$$ is introduced which vanishes at $$\lambda$$ and is a contractive zero divisor in $$\mathbb{A}^ 2$$: $$\| g/f\|_ 2\leq\| g\|_ 2$$ whenever $$g\in\mathbb{A}^ 2$$, $$g(\lambda)=0$$. Further, if $$f\not\equiv 0\in\mathbb{A}^ 2$$, then the infinite product $$B$$, of the $$f_ \lambda$$, as $$\lambda$$ ranges over the zeros of $$f$$, converges. Thus we have $$f=BF$$, with $$F\neq 0$$, $$\| F\|_ 1\leq\| f\|_ 2$$.
The main result, Theorem 2.11, asserts that if a discrete sequence $$\Lambda=\{\lambda_ n\}$$ grows sufficiently slowly, then for almost all independent choices of $$\{\theta_ n\}$$, the sequence $$\{\lambda_ n e^{i\theta_ n}\}$$ is an $$\mathbb{A}^ p$$-zero set, if $$1\leq p\leq 2$$ (for $$p>2$$ slightly less precise conditions are given). The author poses interesting conjectures in §3 concerning the best possible growth allowed of such $$\Lambda$$, and a refinement of a zero-one law. The paper contrasts this work with earlier work of C.Horowitz [Duke Math. J. 41, 693-710 (1974; Zbl 0293.30035)] and H. Hedenmalm [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)]. The Blaschke-type product introduced here differ from those of Hedenmalm that the latter considers an extremal problem over a full sequence $$\Lambda$$, rather than as a product of individual factors.

### MSC:

 30D50 Blaschke products, etc. (MSC2000) 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

### Keywords:

random series; Blaschke product; Bergman spaces

### Citations:

Zbl 0293.30035; Zbl 0734.30040
Full Text:

### References:

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