SOME COMPLETE SIGMA(2)(1) SETS IN HARMONIC-ANALYSIS

We prove that several specific pointsets are complete SIGMA2(1) (compl ete PCA). For example, the class of N0-sets, which is a hereditary cla ss of thin sets that occurs in harmonic analysis, is a pointset in the space of compact subsets of the unit circle; we prove that this point set is complete SIGMA2(1) . We also consider some other aspects of des criptive set theory, such as the nonexistence of Borel (and consistent ly with ZFC, the nonexistence of universally measurable) uniformizing functions for several specific relations. For example, there is no Bor el way (and consistently, no measurable way) to choose for each N0-set , a trigonometric series witnessing that it is an N0-set.