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.
|Number of pages||9|
|Journal||Proceedings of the Indian Academy of Sciences - Mathematical Sciences|
|Publication status||Published - Dec 1990|
- Hausdorff dimension
- Khintchine's theorem
- Metric diophantine approximation