Diophantine Approximation (MATH-540)


The main theme in Diophantine 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.

Timetable and rooms

Lecture: Friday 8:15-10:00 in TBA
Exercises: Friday 10:15-12:00 in TBA


Prof. Friedrich Eisenbrand


Matthieu Haeberle


  • Continued Fractions and convergents
  • Convergents as best approximations
  • Approximation theorems and Liouville’s theorem
  • Quadratic irrational numbers and periodic continued fractions
  • Simultaneous Diophantine approximation
  • Dirichlets Theorems and algorithms
  • Applications of Simultaneous Diophantine approximation in Discrete Optimization
  • Lower bounds based on covering
  • Schmidt’s subspace theorem and open research questions

Required courses

  • Analysis 1 and 2
  • Linear Algebra 1 and 2
  • Rings and Fields

Assessment method

Written exam at the end of the semester.


A. Ya. Khinchin: Continued Fractions

Wolfgang Schmidt: Diophantine Approximation

Some research papers (TBA)