## Roth’s Theorem in the primes

Venue: | Annals of Math |

Citations: | 21 - 4 self |

### BibTeX

@ARTICLE{Green_roth’stheorem,

author = {Ben Green},

title = {Roth’s Theorem in the primes},

journal = {Annals of Math},

year = {},

volume = {161},

pages = {1609--1636}

}

### OpenURL

### Abstract

Abstract. We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1.

### Citations

242 |
Trigonometric series
- Zygmund
- 1988
(Show Context)
Citation Context ...1/2 2 ǫ δ + Cδ N � . x,d We will require several lemmas. The most important is a “discrete majorant property”. Before we state and prove this, we give an elegant argument of Marcinkiewicz and Zygmund =-=[27]-=-. We outline the argument here since we like it and, possibly, it is not particularly well-known. Lemma 6.5 (Marcinkiewicz–Zygmund). Let N be a positive integer, and let f : [N] → C be any function. C... |

228 |
Fourier transformation restriction phenomena for certain lattice subsets and application to the nonlinear evolution equations
- Bourgain
- 1993
(Show Context)
Citation Context ...e restriction theorem for primes was described, in a different context, by Bourgain [4]. Our argument, being visibly analagous to the approach of Tomas, is different and has more in common with §3 of =-=[5]-=-. This more recent paper of Bourgain deals with restriction phenomena of certain sets of lattice points. To deduce Theorem 1.4 from (a variant of) Theorem 1.5 we use a variant of the technique of gran... |

169 |
An introduction to the theory of numbers (Fifth edition
- Hardy, Wright
- 1979
(Show Context)
Citation Context ...(mod q) (q,t)=1 s(mod q) (q,ms+b)=1 e(as/q). = e(−abm/q) � (q,t)=1 = e(−abm/q)µ(q). e(amt/q) This last evaluation, of what is known as a Ramanujan Sum, is well-known and is contained, for example, in =-=[14]-=-. This proves (4.13). Now to obtain σa,q we must simply multiply (4.13) by the factor F = � � 1 − 1 �−1 � � 1 − p 1 � p p<Q p∤m appearing in Lemma 4.6. One gets zero unless (m, q) = 1 and q is Q-smoot... |

142 |
On certain sets of integers
- Roth
- 1954
(Show Context)
Citation Context ...of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound =-=[21]-=- on r3(N), defined 17 years previously by Erdős and Turán to be the density of the largest set A ⊆ [N] containing no non-trivial 3-term arithmetic progression (3AP). Roth was the first person to show ... |

69 |
On Sets of Integers which Contain no Three Terms
- Behrend
- 1946
(Show Context)
Citation Context ...we would like to remark that it is possible, indeed probable, that Roth’s theorem in the primes is true on grounds of density alone. The best known lower bound on r3(N) comes from a result of Behrend =-=[3]-=- from 1946. Proposition 1.6 (Behrend). We have r3(N) � e −C√ log N for some absolute constant C. This may well give the correct order of magnitude for r3(N), and if anything like this could be proved ... |

53 |
On triples in arithmetic progression
- Bourgain
- 1999
(Show Context)
Citation Context ...≪ 1/ log log N. There was no improvement on this bound for nearly 40 years, until Heath-Brown [15] and Szemerédi [22] proved that r3 ≪ (log N) −c for some small positive constant c. Recently Bourgain =-=[6]-=- provided the best bound currently known. Proposition 1.2 (Bourgain). We have r3(N) ≪ (log log N/ log N) 1/2 . The methods of Heath-Brown, Szemerédi and Bourgain may be regarded as (highly non-trivial... |

48 | Integer sets containing no arithmetic progressions
- Heath-Brown
- 1987
(Show Context)
Citation Context ...ct, he proved the following quantitative version of this statement. Proposition 1.1 (Roth). We have r3(N) ≪ 1/ log log N. There was no improvement on this bound for nearly 40 years, until Heath-Brown =-=[15]-=- and Szemerédi [22] proved that r3 ≪ (log N) −c for some small positive constant c. Recently Bourgain [6] provided the best bound currently known. Proposition 1.2 (Bourgain). We have r3(N) ≪ (log log ... |

47 |
A restriction theorem for the Fourier transform, Bull. A.M.S
- Tomas
(Show Context)
Citation Context ...e it, we will establish a somewhat stronger result which we call a restriction theorem for primes. The reason for this is that our argument is very closely analogous to an argument of Tomas and Stein =-=[24]-=- concerning Fourier transforms of measures supported on spheres. A proof of the restriction theorem for primes was described, in a different context, by Bourgain [4]. Our argument, being visibly anala... |

45 |
On certain sets of positive density
- Varnavides
- 1959
(Show Context)
Citation Context ....4. r/∈R � Cδ 1/2 . By (6.6) and Lemma 6.3, a1 behaves a bit like a measure associated to a set of size αN. As promised, we use this information together with an argument originally due to Varnavides =-=[25]-=- to get a lower bound on � a1(x)a1(x + d)a1(x + 2d). Lemma 6.8. For some absolute constant C2 we have � a1(x)a1(x + d)a1(x + 2d) � exp � −C2α −2 log(1/α) � N −1 . x,d∈ZN rsROTH’S THEOREM IN THE PRIMES... |

25 | Restriction and Kakeya phenomena for finite fields - Mockenhaupt, Tao |

21 |
Multiplicative Number Theory, Third edition
- Davenport
- 2000
(Show Context)
Citation Context ...ts the fact that the constant CB is ineffective for any B � 1 due to the possible existence of a Siegel zero. For more information, including a complete proof of Proposition 4.2, see Davenport’s book =-=[7]-=-. The techniques for dealing with the minor arcs are associated with the names of Weyl, Vinogradov and Vaughan. The major arcs. We will have various functons f : [N] → R with �f�∞ = O(log N/N) (4.4) w... |

15 |
On Λ(p)-subsets of squares
- Bourgain
- 1989
(Show Context)
Citation Context ...to an argument of Tomas and Stein [24] concerning Fourier transforms of measures supported on spheres. A proof of the restriction theorem for primes was described, in a different context, by Bourgain =-=[4]-=-. Our argument, being visibly analagous to the approach of Tomas, is different and has more in common with §3 of [5]. This more recent paper of Bourgain deals with restriction phenomena of certain set... |

14 | Restriction theory of the Selberg sieve, with applications
- Green, Tao
(Show Context)
Citation Context ...ntwise. In practise bounds of this latter type will come by restriction theory arguments of the type given in §5. A more general setting for our arguments, along the lines just described, is given in =-=[13]-=-. 7. Acknowledgements The author would like to thank Tim Gowers for his insights into Vinogradov’s threeprimes theorem, which played a substantial part in the development of this paper. He would also ... |

12 |
Bounds in Lebesgue Spaces of Oscillatory Integral Operators. Habilitationsschrift. Univ
- Mockenhaupt
- 1996
(Show Context)
Citation Context ...) = 1). However, there are sets for which (1.1) fails badly when p is not an even integer. For a discussion of this see [10] and for related matters including connections with the Kakeya problem, see =-=[19, 20]-=-. We will apply a variant of Theorem 1.5 for p = 5/2, when it certainly does not seem to be trivial. To prove it, we will establish a somewhat stronger result which we call a restriction theorem for p... |

10 |
Sommes trigonométriques sur les nombres premiers
- Vaughan
- 1977
(Show Context)
Citation Context ...at least when b = m = 1. The first (unconditional) results of this type were obtained by I.M. Vinogradov, and nowadays it is possible to give a rather clean argument thanks to the identity of Vaughan =-=[26]-=-. Chapter 24 of Davenport’s book [7] describes the use of Vaughan’s identity in the more general context of the estimation of sums � n�N Λ(n)f(n). To obtain Lemma 4.9 we used this approach, but could ... |

9 |
Counting sumsets and sum-free sets modulo a prime, Studia Sci
- Green, Ruzsa
(Show Context)
Citation Context ...ttice points. To deduce Theorem 1.4 from (a variant of) Theorem 1.5 we use a variant of the technique of granularization as developed by I.Z. Ruzsa and the author in a series of papers beginning with =-=[9]-=-, as well as a “statistical” version of Roth’s theorem due to Varnavides. We will also require an argument of Marcinkiewicz and Zygmund which allows us to pass from the continuous setting in results s... |

4 |
Exponential sums over primes in an arithmetic progression
- Balog, Perelli
- 1985
(Show Context)
Citation Context ...g. Details may be found in the supplementary document [12]. We remark that existing results in the literature concerning minor arcs estimates for primes restricted to arithmetic progressions, such as =-=[2, 17]-=-, strive for a much better dependence on the parameter m. Lemma 4.10. Suppose that a, q are positive integers with (a, q) = 1, and let θ be a real number such that |θ − a/q| � 1/q 2 . Then λ (Q)∧ b,m,... |

4 | On the Hardy-Littlewood majorant problem
- Green, Ruzsa
(Show Context)
Citation Context ... it is rather straightforward to check that any set does satisfy (1.1) (with C(p) = 1). However, there are sets for which (1.1) fails badly when p is not an even integer. For a discussion of this see =-=[10]-=- and for related matters including connections with the Kakeya problem, see [19, 20]. We will apply a variant of Theorem 1.5 for p = 5/2, when it certainly does not seem to be trivial. To prove it, we... |

2 |
Analytic method of estimates of trigonometric sums by the primes of an arithmetic progression
- Lavrik
- 1979
(Show Context)
Citation Context ...g. Details may be found in the supplementary document [12]. We remark that existing results in the literature concerning minor arcs estimates for primes restricted to arithmetic progressions, such as =-=[2, 17]-=-, strive for a much better dependence on the parameter m. Lemma 4.10. Suppose that a, q are positive integers with (a, q) = 1, and let θ be a real number such that |θ − a/q| � 1/q 2 . Then λ (Q)∧ b,m,... |

1 |
Linear equations
- Balog
- 1992
(Show Context)
Citation Context ...s contain infinitely many 3APs. Van der Corput’s method is very similar to that used by Vinogradov to show that every large odd number is the sum of three primes. Let us also mention a paper of Balog =-=[1]-=- in which it is shown that for any n there are n primes p1, . . .,pn such that all of the averages 1 2 (pi + pj) are prime. In this paper we propose to prove a common generalization of the results of ... |

1 |
Vinogradov’s three-primes theorem, notes available at http://www.dpmms.cam.ac.uk/˜wtg10/3primes.dvi
- Gowers
(Show Context)
Citation Context ...uality then applies. The ingredients are as follows. The almost-primes are eminently suited to applications of sieve techniques. To keep the paper as self-contained as possible, we will follow Gowers =-=[8]-=- and use arguably the simplest sieve, that due to Brun, on both the major and minor arcs. The genuine primes, on the other hand, are harder to deal with. Here we will quote two well-known results from... |

1 |
Restriction and Kakeya phenomena, notes from a course given
- Green
(Show Context)
Citation Context ... –L ∞ estimates �f ∗ ψ ∧ j �∞ ≪ǫ 2 −(1−ǫ)j �f�1 for some ǫ < (p − 2)/2, and also the L 2 –L 2 estimates (2.9) �f ∗ ψ ∧ j �2 2 ≪ǫ ǫj N �f�2. (2.10) Applying the Riesz-Thorin interpolation theorem (see =-=[11]-=-, Chapter 7) will then give �f ∗ ψ ∧ j �p ≪ 2 −δj N −2/p �f�p ′ for some positive δ (depending on ǫ). Summing these estimates from j = 1 to K + 1 will establish (2.7) and hence Theorem 2.1. To define ... |

1 |
Some minor arcs estimates relevant to the paper “Roth’s theorem in the primes”, available at http://www.dpmms.cam.ac.uk/˜bjg23/papers/BG 11 minorarcs.pdf
- Green
(Show Context)
Citation Context ...d this approach, but could afford to obtain results which are rather non-uniform in m due to the restriction m � log N under which we are operating. Details may be found in the supplementary document =-=[12]-=-. We remark that existing results in the literature concerning minor arcs estimates for primes restricted to arithmetic progressions, such as [2, 17], strive for a much better dependence on the parame... |

1 |
The Hardy-Littlewood majorant property for random sets, preprint
- Mockenhaupt, Schlag
(Show Context)
Citation Context ...) = 1). However, there are sets for which (1.1) fails badly when p is not an even integer. For a discussion of this see [10] and for related matters including connections with the Kakeya problem, see =-=[19, 20]-=-. We will apply a variant of Theorem 1.5 for p = 5/2, when it certainly does not seem to be trivial. To prove it, we will establish a somewhat stronger result which we call a restriction theorem for p... |

1 |
Integer sets containing no arithmetic progressions
- Szemerdi
- 1990
(Show Context)
Citation Context ...ollowing quantitative version of this statement. Proposition 1.1 (Roth). We have r3(N) ≪ 1/ log log N. There was no improvement on this bound for nearly 40 years, until Heath-Brown [15] and Szemerédi =-=[22]-=- proved that r3 ≪ (log N) −c for some small positive constant c. Recently Bourgain [6] provided the best bound currently known. Proposition 1.2 (Bourgain). We have r3(N) ≪ (log log N/ log N) 1/2 . The... |

1 |
Notes from a course given at UCLA, available at http://www.math.ucla.edu/˜tao/254b.1.99s/notes1.dvi
- Tao
(Show Context)
Citation Context ...nt by Tomas [24], giving results of a similar nature for spheres in high-dimensional Euclidean spaces, is rather striking. In fact, the reader may care to look at the presentation of Tomas’s proof in =-=[23]-=-, whereupon she will see that there is an almost exact correspondence between the two arguments. To begin with, the proof proceeds by the method of T and T ∗ , a basic technique in functional analysis... |

1 |
Counting sumsets and sum-free sets, preprint
- Green, Ruzsa
(Show Context)
Citation Context ...orted on spheres. To deduce Theorem 4 from (a variant of) Theorem 5 we use a variant of the technique of granularization as developed by I.Z. Ruzsa and the author in a series of papers beginning with =-=[7]-=-, as well as a “statistical” version of Roth’s theorem due to Varnavides. We will also require an argument of Marcinkiewicz and Zygmund which allows us to pass from the continuous setting in results s... |

1 |
On the Hardy-Littlewood majorant problem, in preparation
- Green, Ruzsa
(Show Context)
Citation Context ...eger it is rather straightforward to check that any set does satisfy (1) (with C(p) = 1). However, there are sets for which (1) fails badly when p is not an even integer. For a discussion of this see =-=[8]-=- and for related matters including connections with the Kakeya problem, see [17, 18]. We will apply a variant of Theorem 5 for p = 5/2, when it certainly does not seem to be trivial. To prove it, we w... |

1 |
Roth’s theorem and random and pseudorandom sets, preprint
- Green
(Show Context)
Citation Context ...also require an argument of Marcinkiewicz and Zygmund which allows us to pass from the 2continuous setting in results such as (1) – that is to say, T – to the discrete, namely Z/NZ. In another paper =-=[12]-=- we will prove some results concerning Roth-type theorems in sets which are much sparser, but in a sense more regular, than the primes. Although we have made an effort at clarity in the present paper,... |