  It is well known that a dense subgroup $G$ of the complex unitary group
$U(d)$ cannot be amenable as a discrete group when $d>1$. When $d$ is large
enough we give quantitative versions of this phenomenon in connection with
certain estimates of random Fourier series on the compact group $\bar G$ that
is the closure of $G$. Roughly, we show that if $\bar G$ covers a large enough
part of $U(d)$ in the sense of metric entropy then $G$ cannot be amenable. The
results are all based on a version of a classical theorem of Jordan that says
that if $G$ is finite, or amenable as a discrete group, then $G$ contains an
Abelian subgroup with index $e^{o(d^2)}$.
