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

TL;DR

This paper investigates the problem of finding optimal rational approximations of maximal commutative subgroups of GL(n,R), extending classical Diophantine approximation, with new estimates, algorithms, and geometric interpretations.

Contribution

It introduces the first steps in approximating maximal commutative subgroups of GL(n,R), including new estimates, algorithms, and a geometric framework relating to sails of cones.

Findings

Established approximation estimates for n=2 in real and complex cases.

Developed an algorithm for constructing best approximations of fixed size.

Linked best approximations to geometric structures called sails of cones.

Abstract

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.