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: Oral (session d'hiver)
- Matière examinée: Diophantine approximation
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Automne
- Forme de l'examen: Oral (session d'hiver)
- Matière examinée: Diophantine approximation
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
Semaine de référence
| Lu | Ma | Me | Je | Ve | |
| 8-9 | |||||
| 9-10 | |||||
| 10-11 | |||||
| 11-12 | |||||
| 12-13 | |||||
| 13-14 | |||||
| 14-15 | |||||
| 15-16 | |||||
| 16-17 | |||||
| 17-18 | |||||
| 18-19 | |||||
| 19-20 | |||||
| 20-21 | |||||
| 21-22 |
Légendes:
Cours
Exercice, TP
Projet, autre