Andrej Dujella:

Diophantine approximations and applications

Graduate course     (2011/2012)

Course description

One of the central topics in the part of number theory called diophantine approximations, is the question how well a given irrational number can be approximated by rational numbers. The classical Dirichlet's theorem says that for any irrational number α there exist infinitely many rational numbers p/q such that |α - p/q| < 1/q2. The rational approximations p/q with this property can be obtained using the Farey sequences or continued fractions. Apart from this classical problem, there are some interesting and important modifications, such as asymmetric approximations, simultaneous approximations or approximations by algebraic numbers of given degree.

This course will cover basic results on mentioned problems. It will also cover several applications of results, methods and algorithms from Diophantine approximations, in particular those applications on which the members of the Croatian number theory group have worked in the last few years. They involve applications of Worley's theorem on characterization of good rational approximations in terms of continued fractions in solving certain Diophantine (e.g. Thue) equations, the root separation problem for integer polynomials, and applications of continued fractions and LLL-algorithm in cryptography.

It will be assumed that the students are familiar with the basic notions and results from number theory, at the level covered in the undergraduate course Number Theory.


  1. Y. Bugeaud: Approximation by Algebraic Numbers, Cambridge University Press, Cambridge, 2004.

  2. J. W. S. Cassels: An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, 1965.

  3. D. Hensley: Continued Fractions, World Scientific, Singapore, 2006.

  4. A. Ya. Khinchin: Continued Fractions, Dover, New York, 1997.

  5. S. Lang: Introduction to Diophantine Approximations, Addison-Wesley, Reading, 1966.

  6. P. Q. Nguyen, B. Vallee (Eds.): The LLL Algorithm. Survey and Applications, Springer, Berlin, 2010.

  7. I. Niven: Diophantine Approximations, John Wiley & Sons, New York, 1963.

  8. W. M. Schmidt: Diophantine Approximation, Springer-Verlag, Berlin, 1980, 1996.

  9. W. M. Schmidt: Diophantine Approximation and Diophantine Equations, Springer-Verlag, Berlin, 1991, 1996.

  10. N. P. Smart: The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998.

  11. U. Zannier: Lecture Notes on Diophantine Analysis, Edizioni della Normale, Pisa, 2009.

Lecture notes
(in pdf format; in Croatian)

Seminar topics

Homework exercises:

ex1   ex2   ex3   ex4   ex5  

Some (useful) links

Seminar on Number Theory and Algebra (University of Zagreb)
Number Theory - Undergraduate course (Andrej Dujella)
Cryptography - Undergraduate course (Andrej Dujella)
Software packages of interest to number theory
PARI/GP home page
MAGMA Calculator
Thue equations (Clemens Heuberger)
Diophantine m-tuples page (Andrej Dujella)
Number Theory Web
Number theory groups and seminars
Recommended readings for graduate students in number theory

Andrej Dujella home page