Flows on homogeneous spaces and Diophantine approximation on manifolds

Details Flows on homogeneous spaces and Diophantine approximation on manifolds Flows on homogeneous spaces and Diophantine approximation on manifolds

1998-10-06

arxiv.org/abs/math/9810036v1

Ann. Math. 148 (1998), 339--360.

Mathematics

Number Theory

19 pages. To appear in Annals of Mathematics

Scientific paper

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. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.

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