Simultaneous Diophantine approximation on manifolds and Hausdorff dimension Original Research Article

Let M be an m-dimensional, Ck manifold in image, for any image, and for any τ>0 letimagewhere, for image, image and for image, x =max{x1,…,xn}. In this paper it will be shown that for any Ck manifold M there exist Ck manifolds Mz,Mp arbitrarily ‘Ck-close’ to M with the property that, for all sufficiently large τ,imageThis result shows that the non-zero curvature conditions which have been successfully used to tackle other aspects of the theory of Diophantine approximation on manifolds are unable to distinguish between these two cases when we look at simultaneous Diophantine approximation.