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.