Dynamical Diophantine Approximation II

The proposed project deals with problems that belong to the Geometry of Numbers and to Diophantine Approximation and is conceived as a continuation of the project "Dynamical Diophantine Approximation". An analysis of several applications of Minkowskis Second Convex Body Theorem to the approximation of real numbers by rationals shows that it is necessary to apply the Theorem in a suitable lattice with respect to a family of convex bodies that depend on a parameter. Consequently the successive minima with respect to these convex bodies and the given lattice become functions of this parameter and a lot of information concerning the simultaneous approximation properties of the reals that define the lattice is enclosed in the behaviour of these functions. It is thus quite natural to do Geometry of Numbers for a one-parametric family of convex bodies which could be called "Dynamical Geometry of Numbers". Such a theory should in particular explain the individual and simultaneous behaviour of a given set of successive minima related to this oneparametric family. This could be achieved in a satisfactory way in the three dimensional case and it is very likely that a generalization to n dimensions is possible and will lead to new results in the field of simultaneous approximation. The suggestions of the reviewers of the previous project also go in this direction and encourage further research on the topic. Several more detailed questions of this kind are presented in the following proposal. On the one hand such a theory including all possible applications to Diophantine Approximation represents a desirable goal on its own, on the other hand, once one-parameter problems are well understood, this might lead to new impulse for the study of two-parametric families of convex bodies and the related Littlewood Conjecture.

In the Geometry of Numbers the notion of successive minima of a convex body with respect to a lattice was introduced by Minkowski and has lead to numerous applications in Diophantine Approximation.The idea that the simultaneous approximation properties of a given set of irrational numbers are reected in the dynamical behaviour of the successive minima with respect to a suitable choice of a lattice and a one parametric family of convex bodies originates from previous work and is developed and generalized further in the present project.In fact, the already described behaviour of the three minima functions relative to the simultaneous approximation of two irrational numbers could be generalized to an arbitrary number of irrationals. More precisely, a characteristic description of the graph of the logarithms of the minima functions was achieved that is general enough to be applicable for all simultaneous approximation problems of this kind. With the help of this characterisation it was possible to discover new inequalities between approximation constants and give a geometric, rather elementary proof for them. It can be expected that many more, if not all, simultaneous approximation properties can be derived from the given description.