Geometry of Continued Fractions pp 347-356 | Cite as

# Approximation of Maximal Commutative Subgroups

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## Abstract

We have already discussed some geometric approximation aspects in the plane in Chap. 10: we have studied approximations, first, of an arbitrary ray with vertex at the origin, and second, of an arrangement of two lines passing through the origin. In this chapter we briefly discuss an approximation problem of maximal commutative subgroups of \(\operatorname{GL}(n,\mathbb{R})\) by rational subgroups. In general this problem touches the theory of simultaneous approximation and both subjects of Chap. 10. The problem of approximation of real spectrum maximal commutative subgroups has much in common with the problem of approximation of nondegenerate simplicial cones. So it is clear that multidimensional continued fractions should be a useful tool here. Also we would like to mention that the approximation problem is linked to the so-called limit shape problems.

In this chapter we give general definitions and formulate the problem of best approximations of maximal commutative subgroups. Further, we discuss the connection of three-dimensional maximal commutative subgroup approximation to the classical case of simultaneous approximation of vectors in \(\mathbb{R}^{3}\).

## Keywords

Approximation Problem Simultaneous Approximation Real Spectrum Gaussian Vector Simplicial Cone## References

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