Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

摘要

Let M be an m-dimensional, C~k manifold in R~n, for any k,m,n ∈ N, and for any τ > 0 let y_τ(M) = {x ∈ M:||qx|| < q~(-τ) for infinitely many q ∈ N}, where, for x ∈ R, ||x|| min {|x-i|:i ∈ Z}, and for x = (x_1, …, ||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 τ, dim y_τ(M_z) = 0, dim y_τ(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.