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In Diophantine approximation we want to approximate real numbers by rational numbers. Since the rationals lie dense in the real numbers, we can without any effort find rational numbers arbitrarily close to a given real number. However, it is tempting that the approximating rational number has a small denominator. For instance 355/113 is a much better approximation of π than the naive choice 3141/1000, although the denominator is much smaller. The classical theory of continued fractions gives an effective way to construct "the best possible" approximations of a given real number. Hence, we will start the lectures with an explanation of the theory of continued fractions. Now that we know how the best approximations look like, one might expect that all is done in this direction. This is far from being true! Some natural questions arise, which we aim to answer in this course:
- How good is "best possible"?
- Are there infinitely many "good" approximations to a real number?
- What is it good for?
The second question will be answered by the famous theorem of Roth. Its proof will be a centrepiece of this couse. Concerning the third question, we will try to present two branches of applications of Diophantine approximation. One of these is, of course, the solution of some Diophantine equations (polynomial equations which one wants to solve in the rational integers), as the Thue equation. The other application is a study of transcendental numbers. According to D. Masser, the task of proving that an arbitrarily chosen complex number is transcendental, is a 1:0-problem: It is transcendental with proberbility 1 and one can prove this with proberbility 0. Diophantine approximations provide one of the few tools of proving transcendental results.
Moreover, we will also study approximations by elements in a fixed number field. |
