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

Abstract

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.

14 pages; section 3.3 added

References in corpus (1)

• Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation

Cited by in corpus (1)

• Metric Diophantine approximation for systems of linear forms via dynamics