Hausdorff dimension and generalized simultaneous diophantine approximation

Suppose that m is a positive integer, tau = (tau(1),...,tau(m)) is an element of R-+(m) is a vector of strictly positive numbers, and Q is an infinite set of positive integers. Let W-Q(m; tau) be the set {x is an element of R-m : parallel to x(i)q parallel to < q(-tau i,) 1 less than or equal to i less than or equal to m, for infinitely many q is an element of Q}. In this paper we obtain the Hausdorff dimension of this set. We also consider a generalization of the set W-Q(m; tau), where the error terms q(-tau i) in the inequalities are replaced by psi(i)(q), for general functions psi(i) satisfying a certain condition at infinity.