FWF - Approximation kommutativen Unterg - Maximal Commutative Subgroup Approximation

Project: Research project

Description

In the proposed project we study certain generalizations of concepts from Diophantine approximation. This line of
research was initiated by a recent paper by A.M. Vershik and the author, where we give main notions and
definitions concerning the approximation of maximal commutative subgroups, and where we study the planar case
in more detail. The aim of the current project is to develop efficient methods of approximation of arbitrary lines,
planes, angles, simplicial cones, polygons, polyhedra by rational objects of the same kind (where rationality is
defined in terms of appropriate integer lattices). The approach proposed is based on different methods of
multidimensional continued fractions and their interpretation as maximal commutative subgroups of GL(n,R). The
classical problems of approximation of real numbers by rational numbers as well as simultaneous approximation
are the first steps in this direction. However, at this moment only very little is known about geometric
approximation in more complicated settings.
Within this project we plan to develop an extension of the theory of Markoff minima. In particular we are
interested in the worst approximable subgroups in low-dimensional cases. We will study the distribution of best
approximations related to the structure of the Dirichlet group and geometric properties of cones. The next goal is to
utilize subgroup approximation for the approximation of segments, lines and arrangements of lines, convex
polygons and polyhedra, etc.
J.L. Lagrange, H. Minkowski, C.F. Gauss, and F. Klein discovered that best approximation of real numbers by
rationals is closely related - via continued fractions - to the theory of convex hulls of sets of integer points
contained within angles. The current project will likewise establish connections between approximation problems
and multidimensional continued fractions associated with cones. Further it aims at identifying geometrical objects
to which multidimensional continued fractions can be applied.
In addition we aim to compare approximations generated by the Jacobi-Perron algorithm with maximal
commutative subgroup approximations. The Jacobi-Perron algorithm gives a sequence of approximations of a ray
in space by rational rays. In the two-dimensional case it constructs all best approximations. In the threedimensional
case the situation is not so simple due to the fact that there are empty tetrahedra of arbitrary big
volume. Still, the approximations are relatively good: there are several estimates on their quality. It would be
interesting to compare these approximations with the best approximations in the maximal commutative subgroup'
sense.
This research has connections to other branches of mathematics. In particular it is closely related to approximation
by rational cones which correspond to singularities of complex toric varieties. Another link is to so-called limit
shape problems. The limit shape problems of Young diagrams or convex lattice polygons can be considered in the
simplicial cone setting (instead of its traditional interpretation in the hyperoctant of positive integer points in
space), and in this case the rational approximation of the cone becomes an important argument.
StatusFinished
Effective start/end date1/10/1131/12/13