On the complexity of spectra of bounded analytic functions

Abstract

The spectrum of a bounded analytic function on the unit disk D is the set of the
accumulation points of its zeros. We investigate the computability-theoretic
complexity of spectra of computable bounded analytic functions. While the
spectrum of a bounded analytic function on D is Sigma^0_3--closed, we show the
converse fails. At the same time, we construct a bounded analytic function on D
whose spectrum is Sigma^0_3--complete. We also show that there exists a
Sigma^0_2--closed set of unimodular points which is not the spectrum of any
bounded analytic function on D, while every Pi^0_2--closed set of unimodular
points is. We then turn to uniform Frostman functions. We prove an effective
version of a theorem of Matheson. Namely, every computably closed and nowhere
dense set of unimodular points is the spectrum of a computable uniform Frostman
function.