  We study the distribution of abelian extensions of bounded discriminant of a
number field $k$ which fail the Hasse norm principle. For example, we classify
those finite abelian groups $G$ for which a positive proportion of
$G$-extensions of $k$ fail the Hasse norm principle. We obtain a similar
classification for the failure of weak approximation for the associated norm
one tori. These results involve counting abelian extensions of bounded
discriminant with infinitely many local conditions imposed, which we achieve
using tools from harmonic analysis, building on work of Wright.
