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 non-divergence of unipotent flows on the space of lattices.

Abstract. In this paper we generalize and sharpen D. Sullivan's logarithm law for geodesics by specifying conditions on a sequence of subsets {A t | t ∈ N} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f t of G under which #{t ∈ N | f t x ∈ A t } is infinite for a.e. x ∈ G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan's result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov. §1. Introduction 1.1. This work has been motivated by the following two related results. The first one is the Khinchin-Groshev theorem, one of the cornerstones of metric theory of Diophantine approximation. We will denote by M m,n (R) the space of real matrices with m rows and n columns, and by · the norm on R k , k ∈ N, given by x = max 1≤i≤k |x i |.

Abstract. We study diophantine properties of a typical point with respect to measures on R n . Namely, we identify geometric conditions on a measure µ on R n guaranteeing that µ-almost every y ∈ R n is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called 'friendly'. Examples include smooth measures on nondegenerate manifolds, thus the present paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.

Abstract. Let {g t } be a nonquasiunipotent one-parameter subgroup of a connected semisimple Lie group G without compact factors; we prove that the set of points in a homogeneous space G/Γ (Γ an irreducible lattice in G) with bounded {g t }-trajectories has full Hausdorff dimension. Using this we give necessary and sufficient conditions for this property to hold for any Lie group G and any lattice Γ in G.

Abstract. The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and p-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of R n ) are generalized to the S-arithmetic setting.