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Inventiones mathematicae

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

Logarithm laws for flows on homogeneous spaces

  • D.Y. Kleinbock
  • G.A. Margulis
Article

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 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.

Keywords

Smooth Function Exponential Decay High Rank Homogeneous Space Main Tool 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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: kleinboc@math.rutgers.edu)USA
  2. 2.Department of Mathematics, Yale University, New Haven, CT 06520, USA¶ (e-mail: margulis@math.yale.edu)USA

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