An `almost all versus no' dichotomy in homogeneous dynamics and Diophantine approximation

Details An `almost all versus no' dichotomy in homogeneous dynamics and Diophantine approximation An `almost all versus no' dichotomy in homogeneous dynamics and Diophantine approximation

2009-04-09

arxiv.org/abs/0904.1614v2

Geom. Dedicata 149 (2010), 205-218

Mathematics

Dynamical Systems

14 pages; section 3.3 added

Scientific paper

Let $Y_0$ be a not very well approximable $m\times n$ matrix, and let $M$ be a connected analytic submanifold in the space of $m\times n$ matrices containing $Y_0$. Then almost all $Y\in M$ are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for `quasi-polynomial' flows on on the space of lattices.

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