On Roth type conditions, duality and central Birkhoff sums for i.e.m

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m.) and translation surfaces: the absolute Roth type condition is a weakening of the notion of Roth type i.e.m., while the dual Roth type condition is a condition on the backward rotation number of a translation surface. We show that results on the cohomological equation previously proved in [38] for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted absolute Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with subpolynomial deviations of ergodic averages (corresponding to relative homology classes) distributional limit shapes, which are constructed in a similar way to the limit shapes of Birkhoff sums associated in [36] to functions which correspond to positive Lyapunov exponents.