Diophantine Approximation
Summary
The main theme in Diopahntine 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.
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
- Dirichtets Theorems and algorithms
- Applications of Simultaneous Diophantine approximation in Discrete Optitization
- Lower bounds based on covering
- Schmidt's subspace theorem and open research questions
Learning Prerequisites
Required courses
Analysis 1+2
Linear Algebra 1+2
Rings and Fields
Assessment methods
Written exam at the end of the semester
Resources
Bibliography
A. Ya. Khinchin: Continued Fractions
Wolfgang Schmidt: Diophantine Approximation
Some research papers.
Ressources en bibliothèque
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Diophantine Approximation
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Diophantine Approximation
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks