  We introduce two Diophantine conditions on rotation numbers of interval
exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type
condition} 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} rotation
number of a translation surface. We show that results on the cohomological
equation previously proved in \cite{MY} 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 \emph{absolute} Roth type i.e.m. Under
the dual Roth type condition, we associate to a class of functions with
\emph{subpolynomial} deviations of ergodic averages (corresponding to relative
homology classes) \emph{distributional} limit shapes, which are constructed in
a similar way to the \emph{limit shapes} of Birkhoff sums associated in
\cite{MMY3} to functions which correspond to positive Lyapunov exponents.
