The primes contain arbitrarily long arithmetic progressions

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemerdi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemerdi's theorem that any subset of a sufficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yldrm, which we reproduce here. Using this, one may place (a large fraction of) the primes inside a pseudorandom set of "almost…

• DA
  David and Lucile Packard Foundation
• UD
  Université de Montréal
• UO
  University of British Columbia
• TC
  Trinity College, University of Cambridge
• IO
  Institute of Materials Research and Engineering