DIOPHANTINE APPROXIMATION BY LINEAR-FORMS ON MANIFOLDS

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