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Diophantine vectors in analytic submanifolds of Euclidean spaces

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Abstract

Let \(\mathcal{M}\) be an m-dimensional analytic manifold in ℝn. In this paper, we prove that almost all vectors in \(\mathcal{M}\) (in the sense of Lebesgue measure) are Diophantine if there exists one Diophantine vector in \(\mathcal{M}\).

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References

  1. Pyartli A S. Diophantine approximation on submanifolds of Euclidean space. Funct Anal Appl, 3: 303–306 (1969)

    Article  MATH  Google Scholar 

  2. Bernik V I, Dodson M M. Metric Diophantine Approximation on Manifolds. Cambridge: Cambridge University Press, 1999 (Cambridge Tracts in Math, Vol 137)

    MATH  Google Scholar 

  3. Kleinbock D Y, Margulis G A. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann Math, Ser 2, 148: 339–360 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cheng C Q, Sun Y S. Existence of KAM Tori in degenerate Hamiltonian systems. J Differential Equations, 114: 288–335 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Xu J, You J, Qiu Q. Invariant tori of nearly integrable Hamiltonian systems with degeneracy. Math Z, 226: 375–386 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Sevryuk M B. KAM-stable Hamiltonians, J Dyn Control Sys, 1: 351–366 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Schmidt W M. Metrische sätze über simultane approximation abhängiger grossen. Monatsh Math, 68(2): 154–166 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rüssmann H. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regular Chaotic Dyn, 6: 119–204 (2001)

    Article  MATH  Google Scholar 

  9. Rüssmann H. On twist Hamiltonian. Talk on the Colloque International: Mécanique Céleste et Systémes Hamiltoniens, Marseille, 1990

    Google Scholar 

  10. Starkov A N. Dynamical Systems on Homogeneous Spaces. Providence, RI: Amer Math Soc, 2000 (Transl Math Monograghs, Vol 190)

    Google Scholar 

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Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Rong-mei Cao

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Jian-gong You

Authors
  1. Rong-mei Cao
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  2. Jian-gong You
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Corresponding author

Correspondence to Rong-mei Cao.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant No. 10531050) and the National Basic Research Program of China (Grant No. 2007CB814800)

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Cao, Rm., You, Jg. Diophantine vectors in analytic submanifolds of Euclidean spaces. SCI CHINA SER A 50, 1334–1338 (2007). https://doi.org/10.1007/s11425-007-0088-2

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  • DOI: https://doi.org/10.1007/s11425-007-0088-2

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