We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2 (n+~)/4 best possible. Given a real vector x =(x t , . . . ,x,~-t, 1) ER ~ this CFA generates a sequence of vectors ( ~ ) , . . . , p ~ l , q (k)) EZ n, k = 1, 2, . . . with increasing integers Iq(k)l satisfying for i = 1 , . . . , n 1 '2 Iq(k)]l+ ~-~_~ Ix, p,(~)lq(k) I < 2 (~+2)/' ~ i + x, / By a theorem of Dirichlet this bound is best possible in that the exponent 1 + ~ can in general not be increased.