Green 2008 primes contain arbitrarily long arithmetic progressions

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conjecture_2_2: The paper's statement of Erdős's conjecture on arithmetic progressions, that an infinite sequence of integers with divergent sum of reciprocals contains arbitrarily long arithmetic progressions; the paper notes that it would imply Theorem 1.1 and makes no progress on it.

theorem_1_1: The Green–Tao theorem: for every k the prime numbers contain infinitely many arithmetic progressions of length k, with the remark in Section 11 that the proof gives at least (γ(k)+o(1))N^2/log^k N such progressions below N for some small γ(k) > 0.

theorem_1_2: Szemerédi's theorem in the primes: a set of primes whose relative upper density in the primes is positive contains infinitely many arithmetic progressions of every length; the paper proves Theorem 1.1 in full and sketches in Section 11 the changes that give this theorem.

theorem_3_5: The paper's transference principle: for k ≥ 3 and 0 < δ ≤ 1, a function bounded by a k-pseudorandom measure on Z_N with mean at least δ has k-term progression average at least c(k,δ) − o_{k,δ}(1), with c(k,δ) the constant of Szemerédi's theorem in the form of Proposition 2.3.

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Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 (2008), no. 2, 481-547, doi:10.4007/annals.2008.167.481. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0404188), every other right reserved.

The copy read for this card is the arXiv version stamped "arXiv:math/0404188v6 [math.NT] 23 Sep 2007"; the theorem and page numbers cited here are that version's.

Theorem 1.1 (p. 2) states that for every k there are infinitely many k-term arithmetic progressions consisting of primes, and Theorem 1.2 (p. 2) strengthens this to a Szemeredi theorem in the primes: any set A of primes having positive relative upper density, lim sup pi(N)^{-1}|A cap [1,N]| > 0, contains infinitely many k-term progressions for every k. The paper proves Theorem 1.1 in full; Section 11 (pp. 49-51) states, on pp. 49-50, that the method extends to Theorem 1.2, the one significant change being a residue class b mod W chosen by pigeonhole in place of 1 mod W, and leaves the details to the reader. Three ingredients combine: Szemeredi's theorem, assumed in the form of Proposition 2.3 (p. 4); a new transference principle (Theorem 3.5, pp. 9-10) showing that a positive relative density subset of a sufficiently pseudorandom measure contains long progressions; and the Goldston-Yildirim sieve estimates, reproduced in the paper, which give a pseudorandom measure concentrated on almost primes with respect to which a large fraction of the primes has positive relative density. The argument is ergodic in flavor rather than Fourier analytic, using Gowers-norm-type uniformity and a generalized von Neumann theorem. Section 11 also remarks (p. 49; the introduction says the same on p. 1) that the proof gives at least (gamma(k)+o(1))N^2/log^k N k-term progressions of primes below N for some very small gamma(k) > 0: the order of magnitude of the conjectured Hardy-Littlewood asymptotic C_k N^2/log^k N, but not its constant. The paper also records Erdos's conjecture on arithmetic progressions, that a sequence of integers with divergent reciprocal sum contains arbitrarily long progressions, as Conjecture 2.2 (p. 3), notes that it would imply Theorem 1.1, and makes no progress on it.

Read status: claims checked for Theorems 1.1, 1.2 and 3.5, Conjecture 2.2, Propositions 2.3 and 9.1, Definitions 3.1 to 3.3 and the Section 11 remarks, read clause by clause on the page images; the proofs of Theorem 3.5 and Proposition 9.1 were read for structure only, and the details of Theorem 1.2 that the paper leaves to the reader were not supplied.

Source: https://arxiv.org/abs/math/0404188.

Bears on. #3: Conjecture 2.2 is the problem's question asserted in the affirmative, which the paper records without progress, and Theorem 1.1 is the problem's conclusion for $A$ the set of primes, a special case. #141: Theorem 1.1 gives progressions of primes that need not be consecutive primes, so it does not answer the problem. #219: Theorem 1.1 answers the question yes. #1187: Theorem 1.2 gives the first question's yes once some color class is seen to have positive relative upper density in the primes, a step the paper does not state; the paper says nothing on the second question.

Results.

• Theorem 1.1 (p. 2): for every k there are infinitely many k-term arithmetic progressions of primes; the page also records the quantitative remark of Section 11 (p. 49).
• Theorem 1.2 (p. 2): Szemeredi's theorem in the primes; Section 11 (pp. 49-50) sketches the change to the proof.
• Theorem 3.5 (pp. 9-10): Szemeredi's theorem relative to a k-pseudorandom measure, the paper's transference principle.
• Conjecture 2.2 (p. 3): Erdos's conjecture on arithmetic progressions, recorded without progress.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.