Cauchy transforms and
Beurling-Carleson-Hayman thin sets

Joel H. Shapiro

Michigan Math. J. 27 (1980) 339--351

Abstract: It is proved in this paper that the Carleson sets on the unit circle are precisely the closed subsets of the circle with the property that the distributions they support have Cauchy transforms with bounded characteristic. Elementary facts about Gaussian random series play a role in the proof.