Rational approximation of the maximal commutative subgroups of GL(n,R)

How to find "best rational approximations" of maximal commutative subgroups of GL(n,R)? In this paper we pose and make first steps in the study of this problem. It contains both classical problems of Diophantine and simultaneous approximations as a particular subcases but in general is much wider. We prove estimates for n=2 for both totaly real and complex cases and write the algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.