  We show that a $C^{1+bv}$ circle diffeomorphism with absolutely continuous
derivative and irrational rotation number can be conjugated to diffeomorphisms
that are $C^{1+bv}$ arbitrary close to the corresponding rotation. This
improves a theorem of M.~Herman, who established the same result but starting
with a $C^2$ diffeomorphism. We prove that the same holds for countable Abelian
groups of circle diffeomorphisms acting freely, a result that is new even in
the $C^{\infty}$ context. Related results and examples concerning the
asymptotic distortion of diffeomorphisms are presented. Along this path, we
provide a straightened version of the classical Denjoy-Kocsma inequality for
absolutely continuous potentials.
