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The primes contain arbitrarily long polynomial progressions. (English) Zbl 1230.11018

Acta Math. 201, No. 2, 213-305 (2008); erratum ibid. 210, No. 2, 403-404 (2013).
Let \(\mathcal{A}\) be a subset of the primes having positive relative upper density in the set of all primes. Suppose that one is given polynomials \(f_1(x),\dots,f_k(x)\in\mathbb{Z}[x]\) such that \(f_1(0)= \dots=f_k(0)\). Then it is shown that there are infinitely many pairs of integers \(m\) and \(n\) for which \[ m,\,m+f_1(n),\dots, m+f_k(n)\in\mathcal{A}. \] Taking \(f_j(x)=jx\) one recovers as a special case the theorem of B. Green and T. Tao [Ann. Math. (2) 167, No. 2, 481–547 (2008; Zbl 1191.11025)] concerning arbitrarily long arithmetic progressions in the primes. The theorem should also be compared with the result of A. Sárkőzy [Acta Math. Akad. Sci. Hung. 31, 125–149 (1978; Zbl 0387.10033)] and H. Furstenberg [Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey: Princeton University Press (1981; Zbl 0459.28023)] which shows that if \(f_1(0)=0\) then there are infinitely many pairs of integers \(m\) and \(n\) for which \(m\) and \(m+f_1(n)\) are both prime.
The proof uses the “polynomial Szemerédi theorem” of V. Bergelson and A. Leibman [J. Am. Math. Soc. 9, No. 3, 725–753 (1996; Zbl 0870.11015)], which gives a statement analogous to the main theorem above, but for sets \(\mathcal{A}\) having positive relative upper density in the set of all integers. The overall structure of the argument then follows that in the work of Green and Tao cited above. One proves a quantitative version of Bergelson and Leibman’s polynomial Szemerédi theorem, and then establishes a transference principle, giving a polynomial Szemerédi theorem relative to any suitable pseudorandom measure. Finally an enveloping sieve is used to construct a pseudorandom measure majorizing the characteristic function for the primes.
It should not be imagined however that the present paper is an easy generalization of the Green–Tao argument. Let it suffice to mention just one obstacle that must be overcome. In the linear case, with \(f_j(x)=jx\), one can take the variables \(m\) and \(n\) to be of the same order of magnitude; but now this is impractical. As a result, it is no longer appropriate to use the standard Gowers uniformity norms. Instead the paper introduces a local version of these norms, whose properties the paper develops, and which eventually lead to a local Koopman–von Neumann theorem.

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
11N05 Distribution of primes
11N36 Applications of sieve methods
11K45 Pseudo-random numbers; Monte Carlo methods
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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References:

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