Diophantine approximation and related topics

Distribution modulo 1 and Diophantine Approximation are central fields of study in Number tHeory have a long tradition in particular in Austrian mathematical history. In particular Edmund Hlavka and Wolfgang Schmidt have achieved famous results in these related fields. The planned projekt aims to deal with particular questions concerning these fields that have arised in the last few years. There is special focus on the connection between these fields. A central point is the metric theory of Diophantine approximation, based on Hausdorffs concept of dimension of a very general set, on manifolds. Recently much research on this topic has been done, for example by Beresnevich, Bernik, Bugeaud, Budarina, Dickinson to mention just a few. Another imporant goal is a better understanding of so called Liouville numbers, that have very interesting approximation properties. An open problem posed by Kurt Mahler should be the starting point to link Liouville numbers and the metric theory described above. Another big goal for the project is to obtain better understanding of the distribution of seqeunces modulo 1. Apart from the application in Number Theory, this is of importance for numerical integration, which is of big relevance in natural sciences. In particular the study of certain fast growing sequences modulo 1 and special aspects of biased distributions should be investigated. In this context Pisot numbers are of special interest. The methods in the project will heavily involve the classical results from the so-called geometry of numbers, an elegant combination of Number Theory and Geometry. In particular, Minkowskis lattice point theorems such as central results on continued fractions and their application for rational aproximation should be mentioned.

Diophantine Approximation deals with approximation to real numbers by rational numbers that is fractions. In recent years particularly the metric theory, describing how many real numbers are well or badly ap- proximable in an appropriate sense, has received much attention. This is partly driven by real world applications. Besides Chrytography, especially interference alginment in data science should be emphasized. In the project I have worked on problems in metric Diophantine Approximation, my joint paper in collaboration with M. Hussain (Bendigo, Australien) und D. Simmons (York, UK) roughly speaking deals with metric aspects of rational approximation to hypersurfaces. Furthermore I worked on classical geometry of numbers, a field that interconnects two main areas of mathematics, Number Theory and Geometry, and has long tradition in Austria. A concrete result from my project time are equivalent formulations of Mahlers classification of real numbers, for example I provided a variant based on simultaneous approximation to integral powers of the real number.