Interval exchange transformations (I.E.T.s) are piecewise isometries of an interval to itself, with finitely many discontinuities. They generalize naturally rotations on the circle.
I will present two results on homogeneous diophantine approximations for I.E.T.s. The first result determines the relation between these approximations and the so-called Rauzy-Veech algorithm, a continued fraction procedure for I.E.T.s. The second result generalizes classical Khinchin Theorem to I.E.T.s.
Diophantine conditions for I.E.T.s are strictly related to the Teichmuller flow on strata of abelian differentials. In particular Khinchin Theorem for I.E.T.s implies a sharp estimate on how fast a typical Teichm"uller geodesic wanders towards infinity in its stratum. An interesting corollary is the extension of
Masur's logarithmic law
to strata of abelian differentials.