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Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds

D. Y. Kleinbock and G. A. Margulis
Annals of Mathematics
Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 339-360
Published by: Annals of Mathematics
DOI: 10.2307/120997
Stable URL: http://www.jstor.org/stable/120997
Page Count: 22
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Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds
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Abstract

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindžuk in 1964. We also prove several related hypotheses of Baker and Sprindžuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.

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