Extremal subspaces and their submanifolds

الأصل · EN

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.