MATH-540 / 5 credits

Teacher:

Language: English

Remark: pas donné en 2024-25

## 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

## Required courses

Analysis 1+2

Linear Algebra 1+2

Rings and Fields

## Assessment methods

Written exam at the end of the semester

## Bibliography

A. Ya. Khinchin: Continued Fractions

Wolfgang Schmidt: Diophantine Approximation

Some research papers.

## In the programs

• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Diophantine approximation
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Diophantine approximation
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Diophantine approximation
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Diophantine approximation
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

## Related courses

Results from graphsearch.epfl.ch.