an:05177735
Zbl 1175.11052
Host, Bernard
Arithmetic progressions in prime numbers (after B. Green and T. Tao)
FR
S??minaire Bourbaki. Volume 2004/2005. Expos??s 938--951. Paris: Soci??t?? Math??\-matique de France (ISBN 978-2-85629-224-2/pbk). Ast??risque 307, 229-246, Exp. No. 944 (2006).
2006
a
11N13 11B25 11A41 37A45
primes in arithmetic progression; theorem of Green and Tao; Szemer??di's theorem
In [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{E. Szemer??di} proved that every set of integers of positive upper density contains arbitrarily long arithmetic progressions. Three decades later in [Ann. Math. (2), 167, No. 2, 481--547 (2008; Zbl 1191.11025); preprint \url{http://arxiv1.library.cornell.edu/abs/math/0404188}], \textit{B. Green} and \textit{T. Tao} proved the spectacular result that the sequence of primes contains arbitrarily long arithmetic progressions.
The current article gives a clear exposition in some detail of the ideas, key steps and the main hurdles to be overcome in this proof. The first ingredient is the Green-Tao-Szemer??di theorem, an extension of Szemer??di's theorem, which is stated in terms of a ``pseudo-random measure''. Secondly a suitable pseudo-random measure is established that can be applied to the prime numbers using sieve theory results due to \textit{D. Goldston} and \textit{C. Y. Yildirim} [preprint; \url{http://front.math.ucdavis.edu/math.NT/0504336}]. The author shows how these components combine to establish the result of Green and Tao.
For the entire collection see [Zbl 1105.00003].
Eira J. Scourfield (Egham)
Zbl 0303.10056; Zbl 1191.11025