The primes contain arbitrarily long arithmetic progressions.

*(English)*Zbl 1191.11025This paper needs little introduction: in 2004, the authors proved [arxiv:math/0404188] that the primes contain arbitrarily long arithmetic progressions, a startling result considering that the previous state of the art had been an infinitude of four term arithmetic progressions in which three elements were prime and the fourth a product of at most two primes. As frequently happens when an old problem falls, the solution also precipitated a vast new theory of linear forms in the primes which looks like it will lead to a resolution of the Hardy-Littlewood conjecture for essentially all systems except those describing structures such as twin primes or the Goldbach conjecture. This will be a major achievement, and although the theory has moved on somewhat from this opening of the door, it is still very much worth reading.

The main idea of the paper is a transference principle allowing the authors to transfer results from vanilla structures to pseudo-random versions. They then use some estimates of Goldston and Yıldırım to show that in a suitable sense subsets of the primes behave pseudo-randomly, which allows them to transfer Szemerédi’s theorem to subsets of the primes. A good introduction to this sphere of ideas may be found in the earlier paper of B. Green [Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] where this result is proved for three term progressions.

The main idea of the paper is a transference principle allowing the authors to transfer results from vanilla structures to pseudo-random versions. They then use some estimates of Goldston and Yıldırım to show that in a suitable sense subsets of the primes behave pseudo-randomly, which allows them to transfer Szemerédi’s theorem to subsets of the primes. A good introduction to this sphere of ideas may be found in the earlier paper of B. Green [Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] where this result is proved for three term progressions.

Reviewer: Tom Sanders (Cambridge)