Michel Waldschmidt

Introduction to Diophantine methods: irrationality and transcendence

Content

Diophantine approximation is a chapter in number theory which has witnessed outstanding progress together with a number of deep applications during the recent years. The proofs have long been considered as technically difficult. However, we understand better now the underlying ideas, hence it becomes possible to introduce the basic methods and the fundamental tools in a more clear way.

We start with irrationality proofs. Historically, the first ones concerned irrational algebraic numbers, like the square roots of non square positive integers. Next, the theory of continued fraction expansion provided a very useful tool. Among the first proofs of irrationality for numbers which are now known to be transcendental are the ones by H. Lambert and L. Euler, in the XVIIIth century, for the numbers e and pi. Later, in 1815, J. Fourier gave a simple proof for the irrationality of e.

We first give this proof by Fourier and explain how J. Liouville extended it in 1840 (four years before his outstanding achievement, where he produced the first examples of transcendental numbers). Such arguments are very nice but quite limited, as we shall see. Next we explain how C. Hermite was able in 1873 to go much further by proving the transcendence of the number e. We introduce these new ideas of Hermite in several steps: first we prove the irrationality of the exponential of r for non-zero rational r as well as the irrationality of pi. Next we relate these simple proofs with Hermite's integral formula, following C.L. Siegel (1929 and 1949). Hermite's arguments led to the theory of Padé Approximants. They also enable Lindemann to settle the problem of the quadrature of the circle in 1882, by proving the transcendence of pi.

One of the next important steps in transcendental number theory came with the solution by A.O. Gel'fond and Th. Schneider of the seventh of the 23 problems raised by D. Hilbert at the International Congress of Mathematicians in Paris in 1900: for algebraic alpha and beta with alpha not 0 nor 1 and and beta irrational, the number alpha^beta is transcendental. An example is 2^{\sqrt{2}}, another less obvious example is e^pi. The proofs of Gel'fond and Schneider came after the study, by G. Polya, in 1914, of integer valued entire functions, using interpolation formulae going back to Hermite. We introduce these formulae as well as some variants for meromorphic functions due to R. Lagrange (1935) and recently rehabilitated by T. Rivoal (2006). The end of the course will be devoted to a survey of the most recent irrationality and transcendence results, including results of algebraic independence. We shall also introduce the main conjectures on this topic.

Notes of the course (76 pages, pdf file 596 Ko)
Exercises - first sheet (5 pages, pdf file 152 Ko).
Exercises - second sheet (4 pages, pdf file 160 Ko).
Examen - first session, December 15, 2007 (2 pages + solutions 3 pages, pdf file 180 Ko).
Examen - second session, january 2008 (2 pages + solutions 3 pages, pdf file 180 Ko).

Thursday, September 20, 2007.
Lecture at High School For Gifted Students, Ho Chi Minh City, Arithmetic and secret codes (ppt file 3,4 Mo)

Sunday, September 30, 2007.
Lecture at High School For Gifted Students, Ho Chi Minh City, Coding Theory, Card Tricks and Hat Problems (ppt file 2,2 Mo)