Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in Rⁿ, for any k, m, n ? ?, and for any t> 0 let L_t(M)={x?M : ?qx?<q^-t for infinitely many q??}, where, for x?R, ?x? min{ x - i :i?Z}, and for x=(x₁,?,x_n)???ⁿ, ?X? = max{?x₁pr,?,?x_n?}. In this paper it will be shown that for any C^k manifold M there exist C^k manifolds M_z, M_p arbitrarily 'C^k-close' to M with the property that, for all sufficiently large t, dimLt(M_z)=0, dim L_t(M_p)>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. © 2002 Elsevier Science (USA). All rights reserved.