Multi-dimensional aspects of metric number theory

In this project some specific problems in the mathematical research area of number theory are examined. We will in particular treat the question of how well irrational numbers (such as Pi or the square-root of 2) can be approximated by rational numbers, while keeping the denominator of the approximating rational reasonably small. Applications of this theory can be found in the area of computer-assisted numerical calculations where it is an essential ask to find rational approximations for irrational numbers with as little allocated memory as possible under the requirement that the unavoidable approximation error is of acceptable size. In particular, we will examine in this project specific questions of approximating irrational numbers when the numbers respectively vectors with irrational entries are drawn at random, which will help us to understand the properties of typical irrational numbers. Although this is a topic of fundamental mathematical research, related results have been shown to be applicable in real- life applications such as telecommunication. Several of the questions considered in this project are well-known open conjectures of internationally renowned researchers and it seems impossible to tackle these questions with the already established methods in this area. In this project we will develop innovative tools in order to tackle these open questions. We will use a mixture of techniques from probability theory as well as both classical and innovative tools from analytic and combinatorial number theory. This includes estimates on classical objects of number-theoretic interest such as greatest common divisors and prime numbers.

No additional funding sources recorded.