  Roth type irrational rotation numbers have several equivalent arithmetical
characterizations as well as several equivalent characterizations in terms of
the dynamics of the corresponding circle rotations. In this paper we
investigate how to generalize Roth-like Diophantine conditions to interval
exchange maps. If one considers the dynamics in parameter space one can
introduce two nonequivalent Roth-type conditions, the first (condition (Z)) by
means of the Zorich cocyle, the second (condition (A)) by means of a further
acceleration of the continued fraction algorithm introduced in [10]. A third
very natural condition (condition (D)) arises by considering the distance
between the discontinuity points of the iterates of the map. If one considers
the dynamics of an interval exchange map in phase space then one can introduce
the notion of Diophantine type by considering the asymptotic scaling of return
times pointwise or w.r.t. uniform convergence (resp. condition (R) and (U)). In
the case of circle rotations all the above conditions are equivalent. For
interval exchange maps of three intervals we show that (D) and (A) are
equivalent and imply (Z), (U) and (R) which are equivalent among them. For maps
of four intervals or more we prove several results, the only relation which we
cannot decide is whether (Z) implies (R) or not.
