Dynamical Diophantine Approximation

The proposed project deals with problems that belong to the Geometry of Numbers and to 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 one-parametric family. 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. A second question is concerned with a refinement of W. Schmidts Subspace Theorem due to P. Vojta. Schmidts Theorem says that a certain inequality involving linear forms with algebraic coefficients holds outside of a finite union of proper linear subspaces. Of these subspaces some can be taken to be one dimensional, others are essentially higher dimensional. Vojta showed that the set of these latter subspaces is independent of the parameter e in the inequality. A good quantitative estimate for the cardinality of this subset depending on the dimension would be desirable. For a detailed formulation see the following proposal.

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. Whereas the first and last successive minimum have extensively been studied, investigations on the intermediate successive minima have not been pushed that far. In order to study the simultaneous approximation properties of a given set of irrational numbers, a particular choice of a lattice and of a one parametric family of convex bodies leads to a study of all successive minima as functions of a parameter. In the case where two irrational numbers are to be approximated, it has been possible to give a complete description of the behaviour of all three related successive minima functions with the help of geometric methods. Several inequalities between the maximal and minimal order of magnitude of the successive minima functions are deduced and these can be transformed to well known inequalities between the so called approximation constants. The main advantage of the new approach presented in this project relies on the fact that these inequalities only deal with upper and lower bounds for the successive minima, whereas the geometric method gives a description of the dynamical behaviour of the successive minima functions that encodes all the simultaneous approximation properties of the set of irrational numbers in question.