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SUMMARY:Brian Zilli (Iowa State University)
DTSTART:20240425T180000Z
DTEND:20240425T190000Z
DTSTAMP:20260423T021345Z
UID:OLS/153
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/153/">On
  the spectra of computable bounded analytic functions</a>\nby Brian Zilli 
 (Iowa State University) as part of Online logic seminar\n\n\nAbstract\nMcN
 icholl\, in collaboration with Matheson and later individually\, showed th
 at a Blaschke product is computable if and only if it has a computable zer
 o sequence with computable Blaschke constant. The spectrum of a Blaschke p
 roduct is the set of accumulation points of its zeros. We use Matheson and
  McNicholl's results to consider the arithmetical complexity of such spect
 ra for computable Blaschke products. Namely\, we present results showing t
 hat all such spectra are $\\Sigma^0_3$--closed\, that there exists a $\\Si
 gma^0_3$--complete spectrum\, that every $\\Pi^0_2$--closed subset of the 
 unit circle is a spectrum\, and that there exists a $\\Sigma^0_2$--closed 
 set which is not.\n\n\nWe then turn our attention to uniform Frostman Blas
 chke products\, shown by Frostman to be those with nontangential limits of
  modulus one everywhere (as opposed to generic Blaschke products which\, a
 s inner functions\, are only guaranteed to have radial limits of modulus o
 ne almost everywhere). Matheson showed that the spectra of such functions 
 are precisely the closed\, nonempty\, and nowhere dense subsets of the uni
 t circle. We discuss an effectivization of one direction of his result\, s
 howing that every computably closed\, nonempty\, and nowhere dense subset 
 of the circle is the spectrum of a computable uniform Frostman Blaschke pr
 oduct.\n\nJoint work with Timothy McNicholl\n
LOCATION:https://researchseminars.org/talk/OLS/153/
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