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 |.
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