Diophantine Approximation (DIAMANT)
||8 credit points
||Evertse, J.H. (Universiteit Leiden), Beukers, F. (Universiteit Utrecht)
||Basic knowledge (bachelor level) of algebra and analysis (linear algebra, groups, rings, field extensions, real analysis, some complex analysis).
||This course will be an introduction to irrationality, transcendence and rational approximation to (algebraic) numbers.
In the first part we will discuss classical transcendence results such as the transcendence of pi, Hilbert's 23rd problem, Gel'fond-Baker 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 so-called
Subspace Theorem proved by W.M. Schmidt in 1972,
and give various applications to Diophantine equations
||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.
||Homework assignments and oral examination. More details will be announced during the course.
||Will be announced during the course.