Diophantinc approximation by linear forms on manifolds

M. M. Dodson, B. P. Rynne, J. A G Vickers

Research output: Contribution to journalArticle

Abstract

The following Khintchine-type theorem is proved for manifolds M embedded in R k which satisfy some mild curvature conditions. The inequality |q·x| <?(|q|) where ?(r) ? 0 as r ? 8 has finitely or infinitely many solutions qeZ k for almost all (in induced measure) points x on M according as the sum S r = 1/8 ?(r)r k-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 in M satisfying the inequality infinitely often when ?(r) =r -t . t >k - 1. © 1990 Indian Academy of Sciences.

Original languageEnglish
Pages (from-to)221-229
Number of pages9
JournalProceedings of the Indian Academy of Sciences - Mathematical Sciences
Volume100
Issue number3
DOIs
Publication statusPublished - Dec 1990

Keywords

  • Hausdorff dimension
  • Khintchine's theorem
  • manifolds
  • Metric diophantine approximation

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