FWF - CONTFRA - Continued Fractions and Diophantine Approximation

This research project is concerned with Diophantine approximation, an area of mathematics whose history is thousands of years old. This mathematical area is concerned with the approximation of irrational numbers (that is, numbers which cannot be written as a fraction) by rational numbers (that is, fractions) in a way which is as efficient as possible. Here the word „efficient“ means that the rational approximation should have a denominator which is as small as possible. For example, the number Pi (circle constant) can be approximated very well by the numbers 22/7 and 333/106. According to this description, Diophantine approximation is a part of number theory. However, it has turned out that Diophantine approximation has applications in many other areas of mathematics, and even in other scientific disciplines (such as physics). In this research project, we will investigate some of the many aspects of Diophantine approximation. Among the specific topics are: the investigation of the continued fraction expansion of rational numbers with fixed denominator, but variable numerator; problems in metric Diophantine approximation, connected with the recent scientific breakthrough of Koukoulopoulos-Maynard on the Duffin-Schaeffer conjecture; the construction of sets of very evenly distributed sampling points on a spherical surface. Among the methods used are analysis, number theory, probability theory, and geometry.