Diophantine approximation

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

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: Oral (winter session)
  • Subject examined: Diophantine approximation
  • Courses: 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
  • Courses: 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
  • Courses: 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
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional

Reference week

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