Hausdorff dimension and generalized simultaneous Diophantine approximation

Suppose that m is a positive integer, t = (t₁,...,t_m) ? R^m₊ is a vector of strictly positive numbers, and Q is an infinite set of positive integers. Let W_Q(m;t) be the set {x ? R^m : ?x_iq? < q^¯ti, 1 = i = m, for infinitely many q ? Q}. In this paper we obtain the Hausdorff dimension of this set. We also consider a generalization of the set W_Q(m;t), where the error terms q^¯ti in the inequalities are replaced by ?_i(q), for general functions ?_i satisfying a certain condition at infinity.