Two years ago, during the 26th Brazilian Mathematical Colloquium, I shared a mini-course entitled “Aspectos Ergodicos da Teoria dos Numeros” (Ergodic Aspects of Number Theory) with my two friends Alexander Arbieto and Carlos (Gugu) Moreira. The basic goal of this mini-course was the presentation of several applications of ergodic-theoretical ideas in the solution of number-theoretical problems culminating with a complete proof of the Green-Tao theorem about the existence of arbitrarily long arithmetic progressions of prime numbers. The lecture notes of our talks (written in Portuguese) can be found here.
After this mini-course, Etienne Ghys invited me to write (in English) an expanded version of these lecture notes for a collection of surveys called “Ensaios Matematicos” (Mathematical Essays) edited by Maria Eulalia Vares, Vladas Sidoravicius and Etienne himself.
Of course, I gladly accepted the invitation and the final result of this project can be found here. The working title of the project is “The remarkable effectiveness of Ergodic Theory in Number Theory”. Before describing the content of this survey, let me justify the title by saying that, although it is, of course, inspired by the title “The unreasonable effectiveness of Mathematics in Natural Sciences” of the article of the Nobel laureate Eugene Wigner, I decided to replace the word “unreasonable” by “remarkable” by the following two obvious reasons:
- the interaction between Ergodic Theory and Number Theory is certainly much more recent than the corresponding interaction between Mathematics and Physics, so that it is unfair to say that these interactions have the same status in the mathematical world. In particular, “unreasonable” is not an appropriate word to describe the relationship between Ergodic Theory and Number Theory;
- however, the several successful applications of Ergodic Theory in old problems of Number Theory (such as Margulis solution of Oppenheim conjecture, Elkies-McMullen theorem on the gap distribution of mod 1, Green-Tao theorem about long arithmetic progressions of primes and Einsiedler-Lindenstrauss-Katok partial solution of Littlewood conjecture) allow us to fairly call this interaction “remarkable”.
Now, let me briefly tell you about the content of this book: originally, I intended to discuss the Green-Tao theorem, Elkies-McMullen and Einsiedler-Lindenstrauss-Katok theorem, but the usual space-time problems forced me to choose only 2 theorems. The chosen ones were Green-Tao theorem and Elkies-McMullen theorem by several reasons, but the main fact leading to my choice was that I already had some previous material (in Portuguese) about these two theorems. In any case, as a form of compensation, let me say that I do plan to include some material about Einsiedler-Lindenstrauss-Katok theorem in this blog in a near future.
Finally, let me thank my friend Yuri Lima for a careful revision of a preliminary version of this survey (of course, the remaining mathematical and typographical errors are my entire responsibility).
[Update (August 2, 2009): I’m grateful to Yuri Lima and Jimmy Santamaria for a careful list of corrections and suggestions. The corresponding new version of the survey is now available at the same link.]
[Update (August 18, 2009): I’m thankful to Yuri Lima for a second list of corrections and suggestions concerning the third chapter of the book (namely, Elkies-McMullen theorem). The corresponding new version of the survey is now available at the same link.]
[Update (October 30, 2009): Since the first two chapters of this survey are strongly based on our previous book (in Portuguese), the project was split into two parts: the first one is a joint work with A. Arbieto and C. Gugu Moreira, while the second is my responsibility]