## The primes contain arbitrarily long polynomial progressions

Venue: | Acta Math |

Citations: | 32 - 4 self |

### BibTeX

@ARTICLE{Tao_theprimes,

author = {Terence Tao and Tamar Ziegler},

title = {The primes contain arbitrarily long polynomial progressions},

journal = {Acta Math},

year = {},

pages = {213--305}

}

### OpenURL

### Abstract

Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents

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Citation Context ...édi theorem 69 Appendix C. Elementary convex geometry 71 Appendix D. Counting points of varieties over Fp 73 Appendix E. The distribution of primes 77 References 80 1. Introduction In 1975, Szemerédi =-=[31]-=- proved that any subset A of integers of positive upper |A∩[N]| density limsupN→∞ |[N]| > 0 contains arbitrarily long arithmetic progressions. Throughout this paper [N] denotes the discrete interval [... |

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Citation Context .... . . = Pk(0) = 0 and any ε > 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m), . . . , x+Pk(m) are simultaneously prime. The arguments are based on those in =-=[18]-=-, which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the us... |

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Citation Context ...bitrarily long arithmetic progressions. Throughout this paper [N] denotes the discrete interval [N] := {1, . . .,N}, and |X| denotes the cardinality of a finite set X. Shortly afterwards, Furstenberg =-=[10]-=- gave an ergodic-theory proof of Szemerédi’s theorem. Furstenberg observed that questions about configurations in subsets of positive density in the integers correspond to recurrence questions for set... |

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Citation Context ...3. Unfortunately, as the only known proof of Theorem 1.1 proceeds via infinitary ergodic theory, no explicit bounds are currently known, however it is reasonable to expect (in view of results such as =-=[16]-=-, [34]) that effective bounds will eventually become available.POLYNOMIAL PROGRESSIONS IN PRIMES 13 P1(m ′ ), . . . , x ′ + Pk(m ′ ) ∈ A, x ′ ∈ [N], and m ′ ∈ [M]; more precisely, there are at least ... |

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Citation Context ... open in general (though see [20] for some partial progress in the linear case). 4 Shortly afterwards, the transference principle was also combined in [36] with the multidimensional Szemerédi theorem =-=[12]-=- (or more precisely a hypergraph lemma related to this theorem, see [35]) to establish arbitrarily shaped constellations in the Gaussian primes. A much simpler transference principle is also available... |

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Citation Context ... bound m ≤ x ε for any fixed ε > 0, and thus (by diagonalization) that m = x o(1) ; see Remark 2.4. Our results for the case A = P are consistent with what is predicted by the Bateman-Horn conjecture =-=[3]-=-, which remains totally open in general (though see [20] for some partial progress in the linear case). 4 Shortly afterwards, the transference principle was also combined in [36] with the multidimensi... |

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Citation Context ...r case). 4 Shortly afterwards, the transference principle was also combined in [36] with the multidimensional Szemerédi theorem [12] (or more precisely a hypergraph lemma related to this theorem, see =-=[35]-=-) to establish arbitrarily shaped constellations in the Gaussian primes. A much simpler transference principle is also available for dense subsets of genuinely random sparse sets; see [37].4 TERENCE ... |

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Citation Context ...s an intermediate regime where M = N η0 goes to infinity at a polynomially slower rate than N. In the linear setting, all of these regimes can be equated using the random dilation trick of Varnavides =-=[39]-=-, but this trick is only available in the polynomial setting if one moves to higher dimensions, see Appendix B. 11 The “missing” values of η, such as η1, will be described more fully in Section 2. n∈Y... |

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Citation Context ...D = 1, then the local factor cp(x2 + 1) equals 2/p when p = 1 mod 4 and equals 0 when p = 3 mod4, by quadratic reciprocity. (When p = 2, it is equal to 1/p.) More generally, the Artin reciprocity law =-=[23]-=- relates Artin characters to certain local factors. Deligne’s celebrated proof [7], [8] of the Weil conjectures implies (as a very special case) that cp(P) = 1/p + Ok,D(1/p3/2 ) whenever P ∈ Z[x1, . .... |

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Citation Context ...ortunately, as the only known proof of Theorem 1.1 proceeds via infinitary ergodic theory, no explicit bounds are currently known, however it is reasonable to expect (in view of results such as [16], =-=[34]-=-) that effective bounds will eventually become available.POLYNOMIAL PROGRESSIONS IN PRIMES 13 P1(m ′ ), . . . , x ′ + Pk(m ′ ) ∈ A, x ′ ∈ [N], and m ′ ∈ [M]; more precisely, there are at least 20 cNM... |

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Citation Context ...k of the primes (or any large subset thereof) as a set of positive relative density in the set of almost primes, which (after some application of sieve theory, as in the work of Goldston and Yıldırım =-=[14]-=-) can be shown to exhibit a somewhat pseudorandom behavior. Actually, for technical reasons it is more convenient to work not with the sets of primes and almost primes, but rather with certain normali... |

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Citation Context ...relative to a suitable characteristic factor for the polynomial average being considered. Based on this analogy, and on the description of this characteristic factor in terms of nilsystems (see [22], =-=[24]-=-), one would hope that fU ⊥ could be constructed out of nilsequences. In the case of linear averages this correspondence has already some roots in reality; see [19]. In the special case A = P one can ... |

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Citation Context ... reserving the non-boldface letters for concrete realizations of these indeterminates, which in this paper will always be in the ring of integers Z.14 TERENCE TAO AND TAMAR ZIEGLER Prop 7.4 Prop 7.3 =-=[4]-=- B Th 3.2 Th 4.7 Th 1.3 2 Th 2.3 Th 3.16 4.9 Th 4.5 5,A Th 7.1 Cor 6.4 Prop 6.5 Prop 5.9 7.5 5.10 6.8,A Prop 7.7 Prop 6.2 Prop 5.9 linear case 5.11,5.12,A Th 3.18 Th 12.1 12,C,D,E Cor 11.2 Cor 10.7 11... |

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Citation Context ...ing fact: if Ω is a convex body containing B(0, 1), then Ω can be covered by OD(mes(Ω)) translatesPOLYNOMIAL PROGRESSIONS IN PRIMES 73 of [−1, 1] D . To see this, we use a covering argument of Ruzsa =-=[28]-=-. First observe that because the cube [−1/2, 1/2] D is contained in a dilate of B(0, 1) (and hence Ω) by OD(1), the Minkowski sum Ω + [−1/2, 1/2] D is also contained in an OD(1)dilate of Ω and thus ha... |

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Citation Context ...ateman-Horn conjecture [3], which remains totally open in general (though see [20] for some partial progress in the linear case). 4 Shortly afterwards, the transference principle was also combined in =-=[36]-=- with the multidimensional Szemerédi theorem [12] (or more precisely a hypergraph lemma related to this theorem, see [35]) to establish arbitrarily shaped constellations in the Gaussian primes. A much... |

15 |
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Citation Context ... other out, provided that the algebraic variety {x ∈ F D 1 1 p : WP(x) + b = 0} has the expected density of p + O( p3/2 ) (say) over the finite field affine space F D p . The famous result of Deligne =-=[7]-=-, [8], in which the Weil conjectures were proved, establishes this when WP + b was non-constant and is absolutely irreducible modulo p (i.e. irreducible over the algebraic closure of Fp). However, the... |

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Citation Context ... along these lines. 3 Unlike Szemerédi’s theorem or Sárközy’s theorem, no non-ergodic proof of the BergelsonLeibman theorem in its full generality is currently known. However, in this direction Green =-=[17]-=- has shown by Fourier-analytic methods that any set of integers of positive density contains a triple {x, x + n, x + 2n} where n is a non-zero sum of two squares.POLYNOMIAL PROGRESSIONS IN PRIMES 3 T... |

13 | On sets of natural numbers whose difference set contains no squares
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Citation Context ...eorem for the polynomial P = m 2 providing an upper bound for density of a set A for which A − A does not contain a perfect square. His estimate was later improved by Pintz, Steiger, and Szemerédi in =-=[25]-=-, and then generalized in [2] for P = m k and then [30] for arbitrary P with P(0) = 0. 2 We use Z[m] to denote the space of polynomials of one variable m with integer-valued coefficients; see Section ... |

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Citation Context ...correspond to recurrence questions for sets of positive measure in a probability measure preserving system. This observation is now known as the Furstenberg correspondence principle. In 1978, Sárközy =-=[29]-=- 1 (using the Hardy-Littlewood circle method) and Furstenberg [11] (using the correspondence principle, and ergodic theoretic methods) proved independently that for any polynomial 2 P ∈ Z[m] with P(0)... |

10 |
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Citation Context ...t most Oη4(1) atoms. • (Approximation by continuous functions of G) If A is any atom in Y(G), then there exists a polynomial ΨA : R → R of degree Oη5(1) and coefficients Oη5(1) such that (56) ΨA(x) ∈ =-=[0, 1]-=- for all x ∈ I (57) and ∫ X |1A − ΨA(G)|(ν + 1) = O(η5). Proof. This is essentially [18, Proposition 7.2], but we shall give a complete proof here for the convenience of the reader. We use the probabi... |

9 |
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Citation Context ...e bound 0 ≤ f ≤ ν. This is the third pillar of the argument. The majorant ν acts as an “enveloping sieve” for the primes (or more precisely, for the primes equal to b modulo W), in the sense of [26], =-=[27]-=-. It is defined explicitly in Section 8. However, for the purposes of the proof of the other pillars of the argument (Theorem 3.2 and Theorem 3.16) it will not be necessary to know the precise definit... |

8 |
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(Show Context)
Citation Context ...s properties. We shall construct ν as a truncated divisor sum at level R = Nη2 , although instead of using the Goldston-Yıldırım divisor sum as in [14], [18] we shall use a smoother truncation (as in =-=[32]-=- [21], [20]) as it is slightly easier to estimate14 . The pseudorandomness conditions then reduce, after standard sieve theory manipulations, to the entirely local problem of understanding the pseudor... |

7 | Small gaps between primes - Goldston, Yıldırım |

7 |
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Citation Context ...intwise bound 0 ≤ f ≤ ν. This is the third pillar of the argument. The majorant ν acts as an “enveloping sieve” for the primes (or more precisely, for the primes equal to b modulo W), in the sense of =-=[26]-=-, [27]. It is defined explicitly in Section 8. However, for the purposes of the proof of the other pillars of the argument (Theorem 3.2 and Theorem 3.16) it will not be necessary to know the precise d... |

6 |
Aspects of uniformity in recurrence
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(Show Context)
Citation Context ...rd to deduce Theorem 3.2 from (a multidimensional version of) Theorem 1.1 and the Furstenberg correspondence principle; one can also use the uniform version of the Bergelson-Leibman theorem proved in =-=[5]-=-. As the arguments here are fairly standard, and are unrelated to those in the remainder of the paper, we defer the proof of Theorem 3.2 to Appendix B. 3.3. Pseudorandom measures. To describe the othe... |

6 |
Progressions arithmétiques dans les nombres premiers (d’après B. Green et T
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(Show Context)
Citation Context ...ion f defined in (11). As in all previous sections we are using the notation from Section 2 to define quantities such as W, R, M, b. The measure ν can in fact be described explicitly, following [32], =-=[21]-=-, [20]. Let χ : R → R is a fixed smooth even function which vanishes outside of the interval [−1, 1] and obeys the normalization (71) ∫ 1 0 |χ ′ (t)| 2 dt = 1, but is otherwise arbitrary31 . We then d... |

5 |
An inverse theorem for the Gowers U3 norm, preprint
- Green, Tao
(Show Context)
Citation Context ...r in terms of nilsystems (see [22], [24]), one would hope that fU ⊥ could be constructed out of nilsequences. In the case of linear averages this correspondence has already some roots in reality; see =-=[19]-=-. In the special case A = P one can then hope to use analytic number theory methods to show that fU ⊥ is essentially constant, which would lead to a more precise version of Theorem 1.3 in which one ob... |

5 | Obstructions to uniformity, and arithmetic patterns in the primes, Quart
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(Show Context)
Citation Context ...ater than a large power of R), by sieve theory techniques, and in particular by a method of Goldston and Yıldırım [14], though in the paper here we exploit the smoothness of the cutoff χ (as in [21], =-=[33]-=-, [20]) to avoid the need for multiple contour integration, relying on the somewhat simpler Fourier integral expansion instead. For instance, at such scales it is known from these methods that the ave... |

4 |
conjecture de Weil II, Inst
- La
- 1980
(Show Context)
Citation Context ...r out, provided that the algebraic variety {x ∈ F D 1 1 p : WP(x) + b = 0} has the expected density of p + O( p3/2 ) (say) over the finite field affine space F D p . The famous result of Deligne [7], =-=[8]-=-, in which the Weil conjectures were proved, establishes this when WP + b was non-constant and is absolutely irreducible modulo p (i.e. irreducible over the algebraic closure of Fp). However, there ca... |

3 |
Difference sets without k-th powers
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(Show Context)
Citation Context ... 2 providing an upper bound for density of a set A for which A − A does not contain a perfect square. His estimate was later improved by Pintz, Steiger, and Szemerédi in [25], and then generalized in =-=[2]-=- for P = m k and then [30] for arbitrary P with P(0) = 0. 2 We use Z[m] to denote the space of polynomials of one variable m with integer-valued coefficients; see Section 2 for further notation along ... |

2 |
A polinomial Sárközy-Furstenberg theorem with upper bounds
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(Show Context)
Citation Context ...nd for density of a set A for which A − A does not contain a perfect square. His estimate was later improved by Pintz, Steiger, and Szemerédi in [25], and then generalized in [2] for P = m k and then =-=[30]-=- for arbitrary P with P(0) = 0. 2 We use Z[m] to denote the space of polynomials of one variable m with integer-valued coefficients; see Section 2 for further notation along these lines. 3 Unlike Szem... |

1 |
Convergence of polynomial ergodic averages. Probability in mathematics
- Host, Kra
(Show Context)
Citation Context ... of f relative to a suitable characteristic factor for the polynomial average being considered. Based on this analogy, and on the description of this characteristic factor in terms of nilsystems (see =-=[22]-=-, [24]), one would hope that fU ⊥ could be constructed out of nilsequences. In the case of linear averages this correspondence has already some roots in reality; see [19]. In the special case A = P on... |

1 |
An ergodic transference theorem. Unpublished notes
- Tao
(Show Context)
Citation Context ...heorem, see [35]) to establish arbitrarily shaped constellations in the Gaussian primes. A much simpler transference principle is also available for dense subsets of genuinely random sparse sets; see =-=[37]-=-.4 TERENCE TAO AND TAMAR ZIEGLER Remark 1.5. In view of the generalization of Theorem 1.1 to higher dimensions in [6] it is reasonable to conjecture that an analogous result to Theorem 1.3 also holds... |