# Diophantine approximation (BM)

Vak
2017-2018

## Course description

Basically, Diophantine approximation studies problems such as whether a given number is irrational or transcendental, if irrational, how well it can be approximated by a rational number, and if transcendental, how well it can be approximated by an algebraic number.
Techniques from Diophantine approxiation have been vastly generalized, and with the help of them one can prove that various types of Diophantine equtaions have only finitely many solutions.
In the course we intend to discuss basic approximation theorems due to Dirichlet, transcendence results, application of transcendence techniques to Diophantine equations, approximation of algebraic numbers by rationals, and higher dimensional approximation results.

## Examination

50% homework assignments,
50% oral or written exam (depending on the number of students)

The homework grade is the average of the grades of the homework assignments.
The final grade for the 6EC version of Diophantine approximation is the average of the homework grade and the grade for the (oral or
written) exam, rounded to the closest half, however final grades between 5.5 and 6 are rounded upwards to 6.
There is an additional requirement that the grade for the exam be at least 5.0.
Thus, in case that the grade for the exam is less than 5.0, the final grade for Diophantine approximation will be at most 5, even if the average with the homework grade is 5.5 or higher.

Students who did not get a final grade of at least 6 for Diophantine approximation may try to improve the grade of their exam by doing a resit.
It is not possible to improve the homework grade by submitting other homework assignments. The resit will again be oral or written, depending on the number of students. Again the grade for the resit must be at least 5.0, and the new final grade will be computed in the way described above, where the already existing homework grade is kept, and where the grade of the exam is replaced by that of the resit

Students who want to extend their EC points for Diophantine approximation from 6 to 8 should do first the homework assignments and exam or resit for 6EC. A student who got a final grade of at least 6 for the 6EC version of Diophantine approximation, can obtain the two extra EC by studying some extra material and delivering an extra homework assignment.
The grade for the extra assignment must be at least 6.0.
The grade for the 8EC-version of Diophantine approximation is then computed by taking 75% of the already obtained unrounded final grade for 6EC and 25% of the grade for the extra assignment.

## Prerequisites

Analysis and algebra up to the second year.
We will briefly recall what we need from Algebra 3 (field extensions) so knowledge of that course is not really necessary.

## Literature

Lecture notes will be posted on the course website (under reconstruction).
We will not use the Blackboard learning environment.

## Website

course webpage Diophantine approximation

## Remark 1

The basic course is for 6 ECTS points, but interested students may extend this to 8 points by studying additional material and submitting some extra homework.

## Remark 2

This course will not be given in 2018/19.