Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds

. 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. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk 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. 1. Introduction We start by recalling several basic facts from the theory of simultaneous Diophantine approximation. For x; y 2 R n we let x \Δ y = n X i=1 x i y i ; kxk = max 1in jx i j; \Π(x) = n Y i=1 jx i j and \Π + (x) = n Y i=1 jx i j + ; where jxj + stands for max(jxj; 1). One says that a vector y 2 R n is very well approximable (cf. [S2]), to be abbreviated as VWA, if the following two equivalent conditions are satis...