Extremal subspaces and their submanifolds

By Dmitry Kleinbock. 2002

84 citations recorded by the source corpus.

Abstract. It was proved in the paper [KM1] that the properties of almost all points of $ \mathbb{R}^{n} $ being not very well (multiplicatively) approximable are inherited by nondegenerate in $ \mathbb{R}^{n} $ (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper a.ne 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4316cfd9d35b54edf95be4098c37837a1bcefbc

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Rational points near manifolds and metric Diophantine approximation (2009, background): “The work of Kleinbock and Margulis has also dealt with the far more delicate multiplicative case known as the Baker-Sprindžuk conjecture and led to a surge of activity that led to establishing the extremality of various classes of manifolds and sets; see, for example, [38], [39], [40], [42].”
Badly approximable points on manifolds (2013, background): “Beyond nondegenerate manifolds, it would be interesting to obtain generalisations of Theorems 1 and 2 for friendly measures as defined in [32] as well as for affine subspaces of R and their submanifolds—see [31] for a related context.”
Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds (2002, methodology): “Recently, in a preprint [ Kle02 ] by Kleinbock, using the dynamical approach of [KM98], necessary and sufficient conditions for extremality and strong extremality of any affine subspace of R n have been written down.”
On fractal measures and diophantine approximation (2005, background): “On the other hand, given an affine subspaceL of R n , the paper [ K2 ] exhibits necessary”