In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {At | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements ft of G under which #{t∈ℕ | ftx∈At} 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.
1999-12
2019-04-11T14:34
true
articles
Logarithm laws for flows on homogeneous spaces
https://link.springer.com/10.1007%2Fs002220050350
451-494
research_article
https://scigraph.springernature.com/explorer/license/
1999-12-01
en
doi
10.1007/s002220050350
Mathematical Sciences
readcube_id
4f82683d514ed5eda1bb9f3f748281dca6f4ab972cfcd2be815c1e8ca3046bb6
Inventiones Mathematicae
0020-9910
1432-1297
Pure Mathematics
G.A.
Margulis
138
Kleinbock
D.Y.
Yale University
Department of Mathematics, Yale University, New Haven, CT 06520, USA¶ (e-mail: margulis@math.yale.edu), USA
Springer Nature - SN SciGraph project
3
Rutgers University
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA¶ (e-mail: kleinboc@math.rutgers.edu), USA
pub.1016687602
dimensions_id