Inventiones mathematicae

, Volume 138, Issue 3, pp 451–494 | Cite as

Logarithm laws for flows on homogeneous spaces

  • D.Y. Kleinbock
  • G.A. Margulis


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 xA t } is infinite for a.e. xG/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.


Smooth Function Exponential Decay High Rank Homogeneous Space Main Tool 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • D.Y. Kleinbock
    • 1
  • G.A. Margulis
    • 2
  1. 1.Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA¶ (e-mail:
  2. 2.Department of Mathematics, Yale University, New Haven, CT 06520, USA¶ (e-mail:

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