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
Dans les plans d'études
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Diophantine Approximation
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Diophantine Approximation
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines