Maximal Commutative Subgroup Approximation

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 three- dimensional 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.