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 a n 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 a n element of Q}. In this paper we obtain the Hausdorff dimension of th is set. We also consider a generalization of the set W-Q(m; tau), wher e 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 in finity.