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Proceedings - Mathematical Sciences

, Volume 100, Issue 3, pp 221–229 | Cite as

Diophantinc approximation by linear forms on manifolds

  • M. M. Dodson
  • B. P. Rynne
  • J. A. G. Vickers
Article
  • 12 Downloads

Abstract

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝk which satisfy some mild curvature conditions. The inequality ¦q·x¦ <Ψq¦) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤk for almost all (in induced measure) points x onM according as the sum Σ r =1/∞ Ψ(r)rk−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =rt. τ >k − 1.

Keywords

Metric diophantine approximation Khintchine’s theorem Hausdorff dimension manifolds 

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Copyright information

© Indian Academy of Sciences 1990

Authors and Affiliations

  • M. M. Dodson
    • 1
  • B. P. Rynne
    • 1
    • 2
  • J. A. G. Vickers
    • 1
    • 3
  1. 1.Department of MathematicsUniversity of YorkYorkUK
  2. 2.Department of MathematicsHeriot-Watt UniversityRiccartonUK
  3. 3.Faculty of Mathematical StudiesUniversity of SouthamptonSouthamptonUK

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