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

## TL;DR

This paper introduces new Diophantine conditions on interval exchange maps and translation surfaces, extending previous results on cohomological equations and analyzing distributional limit shapes for functions with subpolynomial deviations.

## Contribution

It defines the absolute and dual Roth type conditions, extending cohomological results and constructing distributional limit shapes for specific classes of functions.

## Key findings

- Extended cohomological equation results to absolute Roth type i.e.m.
- Constructed distributional limit shapes under dual Roth type condition.
- Linked limit shapes to subpolynomial deviations of ergodic averages.

## Abstract

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.