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.