Diophantinc approximation by linear forms on manifolds
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The following Khintchine-type theorem is proved for manifoldsM embedded in ℝk which satisfy some mild curvature conditions. The inequality ¦q·x¦ <Ψ(¦q¦) 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) =r−t. τ >k − 1.
KeywordsMetric diophantine approximation Khintchine’s theorem Hausdorff dimension manifolds
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