Projects offered by Manuel Hauke

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Master thesis:

The area of Diophantine approximation (topic in analytic number theory) aims to find good rational approximations to irrational numbers, with applications both in pure mathematics, but also in real-world applications such as financial mathematics, computer sciences and many more.

Formalized mathematically, one tries to solve the following problem: Given a function \[\psi: \mathbb{N} \to [0,\infty),\] an irrational x and a large integer N, we wish to count the number of solutions to

\[\left\vert x -\frac{p}{q}\right\vert \leq \psi(q), \quad (p,q) \in \mathbb{Z} \times [1,\ldots,N].\]

If x is chosen randomly, then the area of metric Diophantine approximation provides equally elegant and spectacular results such as Khintchine's Theorem or the recently proven Duffin–Schaeffer conjecture, which was one of the reasons James Maynard was awarded with the Fields Medal.

In this project, you will learn a toolbox of methods to tackle questions of this form with the ultimate aim to establish an asymptotic formula (when N → ∞) to questions similar to the one above.

The methods applied are mainly from the area of analytic and combinatorial number theory, combined with a tiny input from probability theory.

Co-supervision: Sigrid Grepstad.