Abstract :
Beginning with an improvement to Dirichletʹs 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{double vertical barα0ndouble vertical bar, double vertical barα1ndouble vertical bar, …, double vertical barαLndouble vertical bar} in the case when α0, α1, …, αL are elements of the vector space image+imageα. 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 Dirichletʹs result. Finally, we close with some remarks on the Littlewood Conjecture in this context.
