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

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Let Y₀ be a not very well approximable m× n matrix, and let M be a connected analytic submanifold in the space of m× n matrices containing Y₀. Then almost all Y∈ 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.