## Random sequences with prescribed radii in the unit ball.(English)Zbl 0865.41007

Given a sequence $$\{a_k \}_k$$ in the unit ball of $$\mathbb{C}^n$$ consider the sequence $$\{a_k (\omega) \}_k$$ obtained by applying to each $$a_k$$ a random rotation. We give conditions on $$\{|a_k |\}_k$$ so that the zero-varieties whose irreducible components are hyperplanes $$X_{a_k (\omega)}= \{z\in \mathbb{B}^n: z\cdot \overline {a}_k (\omega)= |a_k |^2\}$$ are contained in zero sets of Hardy and Bergman functions. Also, we give sharp conditions for $$\{a_k (\omega) \}_k$$ to be interpolating for weighted Bergman spaces almost surely, and for $$\{a_k (\omega) \}_k$$ to be a finite union of separated sequences (in terms of the Gleason distance). Some of these results extend their previous corresponding versions in the disc.

### MSC:

 41A05 Interpolation in approximation theory 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables
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