Ergodic theory and Diophantine approximation for translation surfaces and linear forms
© 2016 IOP Publishing Ltd & London Mathematical Society
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 an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to Schmidt (1960 Can. J. Math. 12 619–31, 1964 Trans. Am. Math. Soc. 110 493–518). 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.
JSA partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 ‘RNMS: GEometric structures And Representation varieties’ (the GEAR Network), and NSF CAREER grant DMS 1351853. JT acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147 and acknowledges support by the Heilbronn Institute for Mathematical Research.
This is the author accepted manuscript. The final version is available from IOP Publishing via the DOI in this record.
Vol. 29, No. 8, pp. 2173 - 2190