Summary
The main theme in Diophantine approximation is to approximate a real number by a rational number with a certain denominator bound. The course covers the case of one real number, that is classical and well understood, and proceeds to simultaneous Diophantine approximations.
Timetable and rooms
Lecture: Friday 8:15-10:00 in TBA
Exercises: Friday 10:15-12:00 in TBA
Lecturer
Assistant
Content
- Continued Fractions and convergents
- Convergents as best approximations
- Approximation theorems and Liouville’s theorem
- Quadratic irrational numbers and periodic continued fractions
- Simultaneous Diophantine approximation
- Dirichlets Theorems and algorithms
- Applications of Simultaneous Diophantine approximation in Discrete Optimization
- Lower bounds based on covering
- Schmidt’s subspace theorem and open research questions
Required courses
- Analysis 1 and 2
- Linear Algebra 1 and 2
- Rings and Fields
Assessment method
Written exam at the end of the semester.
Bibliography
A. Ya. Khinchin: Continued Fractions
Wolfgang Schmidt: Diophantine Approximation