Extremal subspaces and their submanifolds
Abstract
It is known that the properties of almost all points of Rn being not very well (multiplicatively) approximable are inherited by nondegenerate in Rn (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper affine subspaces, and prove that the aforementioned diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional diophantine approximation and dynamics of lattices in Euclidean spaces.
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