We prove that several specific pointsets are complete
Σ
2
1
\Sigma _2^1
(complete PCA). For example, the class of
N
0
{N_0}
-sets, which is a hereditary class 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 pointset is complete
Σ
2
1
\Sigma _2^1
. We also consider some other aspects of descriptive set theory, such as the nonexistence of Borel (and consistently with
ZFC
{\text {ZFC}}
, the nonexistence of universally measurable) uniformizing functions for several specific relations. For example, there is no Borel way (and consistently, no measurable way) to choose for each
N
0
{N_0}
-set, a trigonometric series witnessing that it is an
N
0
{N_0}
-set.