We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses LLL lattice basis reduction. We present a version of the algorithm that runs in polynomial time of the input. MSC2010: primary 11J13; secondary 11Y16, 11J70. 167 168 WIEB BOSMA AND IONICA SMEETS As to the generalization of approximations in higher dimensions Dirichlet [16] proved the following theorem; see Chapter II of [19].