# Mathematics student from Department of Mathematics at Aarhus University wins this year's "Master's thesis prize"

Mathias Løkkegaard Laursen has won Danish Mathematical Society's Master’s thesis prize for 2021. A prize awarded each year to the best mathematics Master's thesis at Danish universities.

"I have always had a great interest in numbers and their infinite properties. Such properties can in some cases be described based on how effectively a number can be approximated by rational numbers. By choosing a criterion for what an 'effective' approximation is, you can thus gather a subset of effectively approximable numbers in the unit interval. I am both very proud and honored to receive the award – especially when it is for something I find as exciting and interesting as mathematics."

– Mathias Løkkegaard Laursen

In his thesis, Mathias begins by providing a detailed exposition of all parts of the highly technical proof from 2020 of the Duffin-Schaeffer conjecture in Diophantine approximation due to Koukoulopoulos and Maynard.

Subsequently he treats a result of Haynes from 2010, which provides a conjecture in the spirit of the Duffin-Schaeffer conjecture, but this time for the p-adic numbers. In addition, Haynes establishes a correspondence between the two conjectures, in the sense that if the p-adic conjecture is true, the original conjecture follows with a few exceptions. Conversely, Haynes proves that the original Duffin-Schaeffer conjecture would imply the p-adic conjecture, provided that the proof of the conjecture is built upon a particular method, namely so-called quasi-independence on average.

According to his supervisor, Simon Kristensen, it is this final result that particularly elevates the Master's thesis. At this point Mathias ends by performing a series of adjustments to the proof of Koukoulopoulos and Maynard, so that quasi-independence on average is brought to light. In doing so, Mathias proves Haynes’ conjecture.

The main result of the thesis has been accepted for publication in Functiones et Approximatio Commentary Mathematici in a joint paper with Mathias' supervisor, Simon Kristensen. In the paper, some further questions are asked, which Mathias has subsequently answered in a follow-up paper.

"It is always a pleasure to supervise good students, and even more so when you as a supervisor just have to guide the student in the direction of a pile of papers with a few loose ideas. When the ideas then come back in a specific form, and the contours of a full research result become clearer and clearer in the process, it is hard to see how the job of a supervisor can be any more satisfying. The process of supervising Mathias was one of these experiences."

– Simon Kristensen, supervisor

The award ceremony will take place 30 September 2022 at 14:00 in Aud D1. There will be a reception afterwards in the Physical Canteen.