  We consider the KAM theory for rotational flows on an $n$-dimensional torus.
We show that if its frequencies are diophantine of type $n-1$, then Moser's KAM
theory with parameters applies to small perturbations of weaker regularity than
$C^n$. Derivatives of order $n$ need not be continuous, but rather $L^2$ in a
certain strong sense. This disproves the long standing conjecture that $C^n$ is
the minimal regularity assumption for KAM to apply in this setting while still
allowing for Herman's $C^{n-\epsilon}\!$-counterexamples.
