Ergodic Theory and Diophantine approximation for translation surfaces and linear forms

Jayadev Athreya, Andrew Parrish, Jimmy Tseng

Published 2014-01-16, updated 2015-09-25Version 2

We derive results on the distribution of directions of saddle connections on translation surfaces, using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give a simple proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt~\cite{SchmidtMetrical}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.