Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

AbstractLet M be an m-dimensional, Ck manifold in Rn, for any k,m,n∈N, and for any τ>0 letSτ(M)={x∈M:||qx||<q−τforinfinitelymanyq∈N},where, for x∈R, ||x||min{|x−i|:i∈Z}, and for x=(x1,…,xn)∈Rn, ||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,dimSτ(Mz)=0,dimSτ(Mp)>0.This 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