
Diophantine Approximation (DIAMANT)
Credits 
8 credit points 
Instructors 
Evertse, J.H. (Universiteit Leiden), Beukers, F. (Universiteit Utrecht) 
Email 
evertse@math.leidenuniv.nl, F.Beukers@math.uu.nl 
Aim 
This course will be an introduction to irrationality, transcendence and rational approximation to (algebraic) numbers. 
Description 
In the first part we will discuss classical transcendence results such as the transcendence of pi, Hilbert's 23rd problem, Gel'fondBaker theory for linear forms in logarithms and applications to diophantine equations.
In addition we give a crash course in algebraic number theoryin order to fully appreciate the results in this area.
In the second part we discuss results concerning the approximation of algebraic numbers by rational numbers.
In 1842, Dirichlet proved an elementary, but now classic theorem, roughly stating that for every real irrational number, there are infinitely many rational numbers approximating it very closely. A central result is a theorem proved by Roth in 1955, implying that for real algebraic numbers, Dirichlet's Theorem is in essence best possible, in the sense that a real algebraic number cannot have rational approximations much better than those provided by Dirichlet's Theorem. We intend to prove a weaker version of Roth's Theorem and give some applications.
Further, we state a far reaching higher dimensional generalization of Roth's Theorem, the socalled Subspace Theorem proved by W.M. Schmidt in 1972, and give various applications to Diophantine equations and inequalities.

Organization 
The basic intention is to give each week a lecture of two hours followed by an exercise session of one hour but we allow some flexibility.

Examination 
Homework assignments and oral examination. More details will be announced during the course. 
Literature 
Will be announced during the course. 
Prerequisites 
Basic knowledge (bachelor level) of algebra and analysis (linear algebra, groups, rings, field extensions, real analysis, some complex analysis). 
