  We determine the structure of a finite subset $A$ of an abelian group given
that $|2A|<3(1-\epsilon)|A|$, $\epsilon>0$; namely, we show that $A$ is
contained either in a "small" one-dimensional coset progression, or in a union
of fewer than $\epsilon^{-1}$ cosets of a finite subgroup.
  The bounds $3(1-\epsilon)|A|$ and $\epsilon^{-1}$ are best possible in the
sense that none of them can be relaxed without tightened another one, and the
estimate obtained for the size of the coset progression containing $A$ is
sharp.
  In the case where the underlying group is infinite cyclic, our result reduces
to the well-known Freiman's $(3n-3)$-theorem; the former thus can be considered
as an extension of the latter onto arbitrary abelian groups, provided that
there is "not too much torsion involved".
