FLOWS ON HOMOGENEOUS SPACES AND DIOPHANTINE APPROXIMATION ON MANIFOLDS

We present a new approach to metric Diophantine approximation on manif olds based on the correspondence between approximation properties of n umbers and orbit properties of certain flows on homogeneous spaces. Th is approach yields a new proof of a conjecture of Mahler, originally s ettled by V. G. Sprindzuk in 1964. We also prove several related hypot heses of Baker and Sprindzuk formulated in 1970s. The core of the proo f is a theorem which generalizes and sharpens earlier results on nondi vergence of unipotent flows on the space of lattices.