  We establish a dichotomy for the rate of the decay of the Ces\`aro averages
of correlations of sufficiently regular functions for typical interval exchange
transformations (IET) which are not rigid rotations (for which weak mixing had
been previously established in the works of Katok-Stepin, Veech, and
Avila-Forni). We show that the rate of decay is either logarithmic or
polynomial, according to whether the IET is of rotation class (i.e., it can be
obtained as the induced map of a rigid rotation) or not. In the latter case, we
also establish that the spectral measures of Lipschitz functions have local
dimension bounded away from zero (by a constant depending only on the number of
intervals). In our approach, upper bounds are obtained through estimates of
twisted Birkhoff sums of Lipschitz functions, while the logarithmic lower
bounds are based on the slow deviation of ergodic averages that govern the
relation between rigid rotations and their induced maps.
