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.
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Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999
A correction to this article is available at http://dx.doi.org/10.1007/s00222-017-0751-3
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Kleinbock, D., Margulis, G. Logarithm laws for flows on homogeneous spaces. Invent. math. 138, 451–494 (1999). https://doi.org/10.1007/s002220050350
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DOI: https://doi.org/10.1007/s002220050350