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We introduce two Diophantine conditions on rotation numbers of intervalexchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth typecondition} is a weakening of the notion of Roth type i.e.m., while the\emph{dual Roth type} condition is a condition on the \emph{backward} rotationnumber of a translation surface. We show that results on the cohomologicalequation previously proved in \cite{MY} for restricted Roth type i.e.m. (on thesolvability under finitely many obstructions and the regularity of thesolutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Underthe dual Roth type condition, we associate to a class of functions with\emph{subpolynomial} deviations of ergodic averages (corresponding to relativehomology classes) \emph{distributional} limit shapes, which are constructed ina similar way to the \emph{limit shapes} of Birkhoff sums associated in\cite{MMY3} to functions which correspond to positive Lyapunov exponents.

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