Beginning with an improvement to Dirichlet's Theorem on simultaneous approximation, in this paper we introduce an alogrithm, in sympathy with the classical continued fraction algorithm, to generate the sequence of best approximates to the system max{‖α 0 n‖, ‖α 1 n‖, …, ‖α L n‖} in the case when α 0 , α 1 , …, α L are elements of the vector space Q+Qα. We then produce best possible upper bounds for the associated diophantine inequality. In addition, we consider implications within the geometry of numbers and investigate the converse of the sharpened version of Dirichlet's result. Finally, we close with some remarks on the Littlewood Conjecture in this context.