## Restriction theory of Selberg’s sieve, with applications, to appear, Journal de Theorie de Nombres de Bordeaux

Citations: | 15 - 7 self |

### BibTeX

@MISC{Green_restrictiontheory,

author = {Ben Green and Terence Tao},

title = {Restriction theory of Selberg’s sieve, with applications, to appear, Journal de Theorie de Nombres de Bordeaux},

year = {}

}

### OpenURL

### Abstract

Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a1,..., ak and b1,...,bk be positive integers. Write h(θ): = ∑ n∈X e(nθ), where X is the set of all n � N such that the numbers a1n + b1,..., akn + bk are all prime. We obtain upper bounds for ‖h ‖ L p (T), p> 2, which are (conditionally on the prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.

### Citations

169 | The primes contain arbitrarily long arithmetic progressions. To appear, Annals of Math. Available at: http://arxiv.org/abs/math/0404188
- Green, Tao
(Show Context)
Citation Context ...any 3-term arithmetic progressions of Chen primes.4 BEN GREEN AND TERENCE TAO Remark. Such a theorem, with 3 replaced by a more general integer k, can probably be obtained by adapting the methods of =-=[18]-=-, in which the primes were established to contain arithmetic progressions of any length k, though the explicit constants obtained by those methods are likely to be substantially worse. Our techniques ... |

149 |
On certain sets of integers
- Roth
- 1953
(Show Context)
Citation Context ...likely to be substantially worse. Our techniques here are somewhat similar in spirit to those in [18] in that they are based on establishing a transference principle, in this case from Roth’s theorem =-=[29]-=- concerning progressions of length three in a dense subset of the integers, to the corresponding result concerning dense subsets of a sufficiently pseudorandom enveloping set. In [18] this transferenc... |

133 |
Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis,” published for
- Montgomery
- 1994
(Show Context)
Citation Context ...e estimate at p = 2 remarked on in the introduction. In this region 2 < p < 4 (or more generally for p /∈ 2N) there are no good monotonicity properties in L p of the Fourier transform to exploit; see =-=[24, p144]-=- for a simple example where monotonicity in this sense breaks down, and [1, 17, 23] for further discussion. On the other hand, all known examples where monotonicity breaks down are rather pathological... |

116 | A new proof of Szemeredi’s theorem for arithmetic progressions of length four
- Gowers
- 1998
(Show Context)
Citation Context ...guments here only apply to progressions of length 3. Indeed, it is by now wellunderstood that traditional harmonic analysis arguments cannot suffice to deal with progressions of length 4 or more (see =-=[14]-=- for a further discussion). We begin by recalling a formulation of Roth’s theorem [29] due to Varnavides [35]. Theorem 5.1. [29, 35] Let 0 < δ � 1 be a positive numbers, and let N be a large prime par... |

52 |
A restriction theorem for the Fourier transform
- Tomas
(Show Context)
Citation Context ... applications (indeed, Theorem 1.1 will be an almost immediate consequence of it). As in many arguments in restriction theory, the argument is based in spirit on the Tomas-Stein method, first used in =-=[34]-=-. Proposition 4.1 (Lq Restriction estimate for β). Let R, N be large numbers such that 1 ≪ R ≪ N1/10 and let k, F, βR be as in Proposition 3.1. Let f : ZN → C be arbitrary. Then for every 1 < q < 2 we... |

50 |
On certain sets of positive density
- Varnavides
- 1959
(Show Context)
Citation Context ...monic analysis arguments cannot suffice to deal with progressions of length 4 or more (see [14] for a further discussion). We begin by recalling a formulation of Roth’s theorem [29] due to Varnavides =-=[35]-=-. Theorem 5.1. [29, 35] Let 0 < δ � 1 be a positive numbers, and let N be a large prime parameter. Suppose that f : ZN → R + is a function satisfying the uniform bounds 0 � f(n) � 1 for all n ∈ ZN12 ... |

45 |
On the representation of a large even integer as the sum of a prime and the product of at most two primes
- Chen, Wang
- 1989
(Show Context)
Citation Context ...m 1.2, which stated that there are infinitely many 3-term progressions of Chen primes. A Chen prime, recall, is a prime p such that p + 2 is a product of at most two primes, and Chen’s famous theorem =-=[6]-=- is that there are infinitely many such primes. Chen actually proved a somewhat stronger result, in which the smallest prime factor of p+2 can be bounded below by p γ . It is important for us to have ... |

31 | Higher correlations of divisor sums related to primes III: small gaps between primes
- Goldston, Yıldırım
- 2007
(Show Context)
Citation Context ... with βR, we write this majorant in the form where β ′ R p (n) = G(R)(∑ d|n d�R λ GY d ) 2 (8.7) λ GY µ(d) log(R/d) d = . log R Objects of this type were extensively analysed by Goldston and Yıldırım =-=[11, 12, 13]-=-, and in particular they saw how to asymptotically evaluate certain correlations of the form (8.1) provided R � Ncm . This was a crucial ingredient in our work in [18], but we should remark that other... |

22 | Recent progress on the restriction conjecture, Park City notes
- Tao
(Show Context)
Citation Context ...erties it is often possible to recover estimates of Hardy-Littlewood majorant type. This is known as the restriction phenomenon and has been intensively studied in harmonic analysis; see for instance =-=[32]-=- for a recent survey of this theory. The application of ideas from restriction theory to number theory was initiated by Bourgain in the papers [3, 4] (among others). In [3], which is of the most relev... |

17 |
The polynomial X 2 +Y 4 captures its primes
- Friedlander, Iwaniec
- 1998
(Show Context)
Citation Context ...f pursuing the matter simply to obtain correlation estimates for βR. The function HR is of some interest in its own right, however, and had an auxilliary role for instance in the ground-breaking work =-=[9]-=-. Let us conclude with the following remark. As far as enveloping sieves for the primes are concerned, it seems that both βR and β ′ R have their own role to play. βR is useful if one wants to do harm... |

17 |
On the Barban-Davenport-Halberstam theorem
- Hooley
- 1976
(Show Context)
Citation Context ...ark that other aspects of the function β ′ R were investigated much earlier, and indeed β ′ R was known to Selberg. It has also featured in works of Friedlander-Goldston [8], Goldston [10] and Hooley =-=[21]-=-, among others. The consideration of the weights λGYsRESTRICTION THEORY OF THE SELBERG SIEVE, WITH APPLICATIONS 27 and the associated majorant β ′ R may be motivated by looking at an asymptotic for th... |

16 |
transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
- Fourier
- 1993
(Show Context)
Citation Context ...ively studied in harmonic analysis; see for instance [32] for a recent survey of this theory. The application of ideas from restriction theory to number theory was initiated by Bourgain in the papers =-=[3, 4]-=- (among others). In [3], which is of the most relevance to us, it is shown how to obtain bounds for ‖ ˆ f‖p, where f is a function supported on the primes. Such a bound was also obtained in a paper of... |

15 |
On Λ(p)-subsets of squares
- Bourgain
- 1989
(Show Context)
Citation Context ...ively studied in harmonic analysis; see for instance [32] for a recent survey of this theory. The application of ideas from restriction theory to number theory was initiated by Bourgain in the papers =-=[3, 4]-=- (among others). In [3], which is of the most relevance to us, it is shown how to obtain bounds for ‖ ˆ f‖p, where f is a function supported on the primes. Such a bound was also obtained in a paper of... |

12 |
Small gaps between primes, I, preprint
- Goldston, Yıldırım
(Show Context)
Citation Context ... with βR, we write this majorant in the form where β ′ R p (n) = G(R)(∑ d|n d�R λ GY d ) 2 (8.7) λ GY µ(d) log(R/d) d = . log R Objects of this type were extensively analysed by Goldston and Yıldırım =-=[11, 12, 13]-=-, and in particular they saw how to asymptotically evaluate certain correlations of the form (8.1) provided R � Ncm . This was a crucial ingredient in our work in [18], but we should remark that other... |

12 |
Trigonometric series I, 2nd Ed
- Zygmund
- 1959
(Show Context)
Citation Context ...k E ( |an| 2 βR(n)|1 � n � N ) p/2 b∈ZN from (4.6) simply by multiplying each coefficient an by eN(θn). Integrating this in θ from 0 to 1 gives (4.7). We remark that the Marcinkiewicz-Zygmund theorem =-=[38]-=- allows one to obtain (4.6) from (4.7); see [16] for further discussion. The utility of Proposition 4.2 is markedly increased by the next lemma. Lemma 4.3. Suppose that R � √ N. Then E(βR(n)|1 � n � N... |

11 |
A lower bound for the second moment of primes in short intervals
- Goldston
- 1995
(Show Context)
Citation Context ...ut we should remark that other aspects of the function β ′ R were investigated much earlier, and indeed β ′ R was known to Selberg. It has also featured in works of Friedlander-Goldston [8], Goldston =-=[10]-=- and Hooley [21], among others. The consideration of the weights λGYsRESTRICTION THEORY OF THE SELBERG SIEVE, WITH APPLICATIONS 27 and the associated majorant β ′ R may be motivated by looking at an a... |

11 | The Hardy-Littlewood Method, 2nd ed. Cambridge Tracts - Vaughan - 1997 |

9 |
An asymptotic estimate related to Selberg’s sieve
- Graham
- 1978
(Show Context)
Citation Context ...t βR and β ′ R are, for many purposes, rather similar. One way to see this is to inspect the quadratic form Q(λGY ). The asymptotic Q(λ GY ) = 1 log R 1 + O( log 2 R ) follows from a result of Graham =-=[15]-=- and can easily be proved using Goldston and Yıldırım’s method (as exposited, for example, in [18, Ch. 9]). Comparing with (8.5) and using the asymptotic (8.6) one sees that ∑ δ�R φ(δ)(u GY δ − uSEL δ... |

9 |
Additive properties of dense subsets of sifted sequences
- Ramaré, Ruzsa
(Show Context)
Citation Context ...(R) on XR!. We now give a Fourier representation for βR similar to (7.6), but with slightly larger Fourier coefficients.RESTRICTION THEORY OF THE SELBERG SIEVE, WITH APPLICATIONS 21 Proposition 7.6. =-=[28]-=- We have the representation βR(n) = ∑ ∑ w(a/q)eq(−an), q�R2 a∈Z∗ q where the coefficients w(a/q) obey the bounds |w(a/q)| � 3 ̟(q) |s(a/q)| (7.11) (here ̟(q) denotes the number of prime factors of q).... |

7 |
On Snirel’man’s constant, Ann
- Ramaré
- 1995
(Show Context)
Citation Context ... the bounds |w(a/q)| � 3 ̟(q) |s(a/q)| (7.11) (here ̟(q) denotes the number of prime factors of q). Also we have w(0) = 1. Remark. More precise asymptotics for w(a/q) are available, see [28] (and see =-=[27]-=- for an even more precise statement in the case F(n) = n). We will not need these refinements here, however. Proof. We follow the arguments of Ramaré and Ruzsa [28]. Our starting point is the formula ... |

6 | Variance of distribution of primes in residue classes
- Friedlander, Goldston
- 1996
(Show Context)
Citation Context ...ork in [18], but we should remark that other aspects of the function β ′ R were investigated much earlier, and indeed β ′ R was known to Selberg. It has also featured in works of Friedlander-Goldston =-=[8]-=-, Goldston [10] and Hooley [21], among others. The consideration of the weights λGYsRESTRICTION THEORY OF THE SELBERG SIEVE, WITH APPLICATIONS 27 and the associated majorant β ′ R may be motivated by ... |

5 |
On an elementary method in the theory of primes
- Selberg
- 1947
(Show Context)
Citation Context ... a ∈ Z∗ q . If q is not square-free, then w(a/q) = 0. Similarly, if γ(q) = 1 and q > 1, then w(a/q) = 0. Remark. The enveloping sieve βR is essentially a normalised version of the one used by Selberg =-=[31]-=- in obtaining the upper bound (1.8). It will be constructed quite explicitly in §7, where a full proof of Proposition 3.1 will also be supplied. One does not need to know the exact construction in ord... |

5 |
Some series involving Euler’s function
- Ward
- 1927
(Show Context)
Citation Context ...t G(R) ∼ log R. In fact one has the asymptotic G(R) = log R + γ + ∑ log p + o(1). (8.6) p(p − 1) This may be found in [27], for example; Montgomery [25] traces the result back at least as far as Ward =-=[37]-=-. We remark that all this together with Lemma 7.9 (which, in our new notation, tells us that |λSEL d | � 1) allows us to recover the fact that if R = N1/2−ǫ then ∑ βR(n) = N(1 + oǫ(1)). n�N This also ... |

4 |
On the upper and lower majorant properties in L p
- Bachelis
- 1973
(Show Context)
Citation Context ...more generally for p /∈ 2N) there are no good monotonicity properties in L p of the Fourier transform to exploit; see [24, p144] for a simple example where monotonicity in this sense breaks down, and =-=[1, 17, 23]-=- for further discussion. On the other hand, all known examples where monotonicity breaks down are rather pathological. When the majorant βR enjoys additional Fourier or geometric properties it is ofte... |

4 |
The prime k-tuples conjecture on average, Analytic Number Theory (Allerton Park
- Balog
- 1989
(Show Context)
Citation Context ...known to be non-empty. Thus there is certainly no hope of obtaining an asymptotic formula for h on the major arcs, unless one is willing to obtain results on average over choices of F, as was done in =-=[2]-=-. On the other hand, the Hardy-Littlewood k-tuple conjecture asserts that for a fixed F N hN;F(0) = |X(F) ∩ [1, N]| = (1 + oF(1))SF log k N as N → ∞, where SF is the singular series SF := ∏ p (1.6) γ(... |

4 |
Sur une somme liée à la fonction Möbius
- DRESS, IWANIEC, et al.
- 1983
(Show Context)
Citation Context ...rt might require one to address the function HR(n) := ∑ µ(d), concerning oneself in particular with the 2m th moment of this function. When m = 1 this was investigated by Dress, Iwaniec and Tenenbaum =-=[7]-=- and when m = 2 by Motohashi [26]. At this point there are already some rather thorny issues involved, and it does not seem worth the effort of pursuing the matter simply to obtain correlation estimat... |

4 | On the Hardy-Littlewood majorant problem
- Green, Ruzsa
(Show Context)
Citation Context ...more generally for p /∈ 2N) there are no good monotonicity properties in L p of the Fourier transform to exploit; see [24, p144] for a simple example where monotonicity in this sense breaks down, and =-=[1, 17, 23]-=- for further discussion. On the other hand, all known examples where monotonicity breaks down are rather pathological. When the majorant βR enjoys additional Fourier or geometric properties it is ofte... |

3 |
Sieve methods, Graduate course
- Iwaniec
- 1996
(Show Context)
Citation Context ...tually proved a somewhat stronger result, in which the smallest prime factor of p+2 can be bounded below by p γ . It is important for us to have this extra information. In Iwaniec’s unpublished notes =-=[22]-=- one may find a proof of Chen’s theorem which leads to the value γ = 3/11. For the purposes of this section we will use the term Chen prime to refer to a p for which p + 2 is either prime or a product... |

3 |
Selberg’s work on the zeta-function, in Number Theory, Trace Formulas and Discrete Groups (a Symposium in Honor of Atle Selberg
- Montgomery
- 1987
(Show Context)
Citation Context ...to the choice λd = λSEL d . Now it is well-known that G(R) ∼ log R. In fact one has the asymptotic G(R) = log R + γ + ∑ log p + o(1). (8.6) p(p − 1) This may be found in [27], for example; Montgomery =-=[25]-=- traces the result back at least as far as Ward [37]. We remark that all this together with Lemma 7.9 (which, in our new notation, tells us that |λSEL d | � 1) allows us to recover the fact that if R ... |

3 |
On an additive property of squares and primes, Acta Arithmetica 49
- Ruzsa
- 1988
(Show Context)
Citation Context ...2 2+ǫ |1 � n � N b∈B d|B|(n) 2+ǫ ǫ ) ǫ 2+ǫ |1 � n � N . � (|B| 1+ǫ ) 2 ǫ 1+ 2+ǫ = |B| 2+ǫ. On the other hand, from standard moment estimates for the restricted divisor function d|B| (see for instance =-=[3, 30]-=-) we have E(d|B|(n) m |1 � n � N) ≪ǫ,m |B| ǫ for any m � 1 (in fact one can replace |B| ǫ with log 2m−1 |B| when m is an integer - see [30]). Applying this with m := 2+ǫ ǫ and working back, one gets (... |

2 |
Applications of sieve methods in number theory, Cambridge Tracts
- Hooley
- 1976
(Show Context)
Citation Context ...bounds but as an interesting function in its own right is one we got by reading papers of Ramaré [27] and Ramaré-Ruzsa [28]. Ramaré generously acknowledges that the essential idea goes back to Hooley =-=[20]-=-. It is in those papers (and in fact so far as we know only in those papers) where one finds the term enveloping sieve. The name, though perhaps nonstandard, seems to us to be appropriate for describi... |

2 |
A multiple sum involving the Möbius function, preprint. Available at http://www.arxiv.org/pdf/math.NT/0310064
- Motohashi
(Show Context)
Citation Context ...the function HR(n) := ∑ µ(d), concerning oneself in particular with the 2m th moment of this function. When m = 1 this was investigated by Dress, Iwaniec and Tenenbaum [7] and when m = 2 by Motohashi =-=[26]-=-. At this point there are already some rather thorny issues involved, and it does not seem worth the effort of pursuing the matter simply to obtain correlation estimates for βR. The function HR is of ... |

2 |
Arithmetic progressions of prime-almost-prime twins
- Tolev
- 1999
(Show Context)
Citation Context ... under control). Remark. We note that results along the lines of Theorem 1.2 can be approached using sieve methods and the Hardy-Littlewood circle method in a more classical guise. For example, Tolev =-=[33]-=- showed that there are infinitely many 3-term progressions p1 < p2 < p3 of primes such that pi + 2 is a product of at most ri primes, where (r1, r2, r3) can be taken to be (5, 5, 8) or (4, 5, 11). 2. ... |

1 |
Roth’s theorem in the primes, preprint. Available at: http://www.arxiv.org/pdf/math.NT/0302311
- Green
(Show Context)
Citation Context ... X(F) is replaced by other similar sets, for instance if primes are replaced by almost primes, or with any subset of the almost primes. Combining this observation with an argument of the first author =-=[16]-=-, we prove the following result. Theorem 1.2. Define a Chen prime to be a prime p such that p + 2 is either prime or a product of two primes. Then there are infinitely many 3-term arithmetic progressi... |