## Arithmetic progressions and the primes - El Escorial lectures

Venue: | Collectanea Mathematica (2006), Vol. Extra., 37-88 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Tao_arithmeticprogressions,

author = {Terence Tao},

title = {Arithmetic progressions and the primes - El Escorial lectures},

booktitle = {Collectanea Mathematica (2006), Vol. Extra., 37-88 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial},

year = {}

}

### OpenURL

### Abstract

Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.

### Citations

150 | The primes contain arbitrarily long arithmetic progressions
- Green, Tao
(Show Context)
Citation Context ...r relatively dense subsets of primes, but relatively dense subsets of almost primes (numbers containing no small prime factors); we shall return to this point later. In 2004, Ben Green and the author =-=[23]-=- were able to extend this theorem to arbitrarily long progressions, by replacing Fourier-analytic ideas with ergodic theory ones: Theorem 1.5. [24] Let A ⊂ P be a subset of primes with positive relati... |

138 |
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
- Furstenberg
- 1977
(Show Context)
Citation Context ...h theory) and very complicated. A substantially shorter proof - but one involving the full machinery of measure theory and ergodic theory, as well as the axiom of choice - was obtained by Furstenberg =-=[10]-=-, [11] in 1977. Since then, there have been two other types of proofs; a proof of Gowers [16], [17] in 2001 which combines “higher order” Fourier analytic methods with techniques from additive combina... |

137 |
On certain sets of integers
- Roth
- 1953
(Show Context)
Citation Context ...initely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. Introduction A celebrated theorem of Roth =-=[36]-=- in 1953 asserts: Theorem 1.1 (Roth’s theorem, first version). [36] Let A ⊂ Z + be a subset of inte1 gers with positive upper density, thus lim supN→∞ |A ∩ [1, N]| > 0. Then A contains N infinitely ma... |

133 |
Some problems on partitio numerorum III : On the expression of a number as a sum of primes
- Littlewood
- 1923
(Show Context)
Citation Context ...by the statement lim inf N→∞ E(Λ(n)Λ(n + 2) : 1 � n � N) > 0. is non-zero for infinitely many N. In fact Hardy and Littlewood made the stronger conjecture, the Hardy-Littlewood prime tuple conjecture =-=[26]-=-, which would imply the twin prime conjecture, and would indeed verify the stronger estimate E(ΛN(n)ΛN(n + 2) : 1 � n � N) = B2 + o(1) where B2 is the Twin prime constant B2 := ∏ P(n, n + 2 coprime to... |

123 |
der Waerden. Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wiskunde
- van
- 1927
(Show Context)
Citation Context ...orem is the most difficult component of the argument; it uses the uniform almost periodicity control on ˜g to “color” the orbit of T n ˜g and hence T n g, and then invokes the van der Waerden theorem =-=[44]-=- to extract arithmetic progressions from g. As such, this part of the argument can be considered to be more combinatorial than ergodic or analytic in nature. 5. Progressions in the primes There are ma... |

106 | A new proof for Szemeredi’s theorem for arithmetic progressions of length four. Geometric and Functional Analysis
- Gowers
- 1998
(Show Context)
Citation Context ...chinery of measure theory and ergodic theory, as well as the axiom of choice - was obtained by Furstenberg [10], [11] in 1977. Since then, there have been two other types of proofs; a proof of Gowers =-=[16]-=-, [17] in 2001 which combines “higher order” Fourier analytic methods with techniques from additive combinatorics; and also arguments of Gowers [18] and Rodl-Skokan [34], [35] using the machinery of h... |

82 | On the density of some sequences of integers - Erdős - 1948 |

79 | Polynomial extensions of van der Waerden’s and Szemerédi’s theorems - Bergelson, Leibman - 1996 |

66 | Regularity lemma for k-uniform hypergraphs
- Rödl, Skokan
(Show Context)
Citation Context ...s of proofs; a proof of Gowers [16], [17] in 2001 which combines “higher order” Fourier analytic methods with techniques from additive combinatorics; and also arguments of Gowers [18] and Rodl-Skokan =-=[34]-=-, [35] using the machinery of hypergraphs. While we will not discuss all these separate proofs in detail here, we will need to discuss certain ideas from each of these arguments as they will eventuall... |

64 | The Theory of the Riemann Zeta Function - Titchmarsh - 1986 |

55 | L p estimates on the bilinear Hilbert transform for 2
- Lacey, Thiele
- 1997
(Show Context)
Citation Context ...emark 2.4. Interestingly, estimates of this type (after being suitably localized in phase space) have proven to be crucial in recent progress in understanding the bilinear Hilbert transform (see e.g. =-=[30]-=-), or at least in understanding the contribution of individual “trees” to that transform. Indeed there is some formal similarity between the trilinear form Λ3 and the trilinear form Λ(f, g, h) := p.v.... |

51 |
On sets of integers containing no four elements in arithmetic progression
- Szemerédi
- 1969
(Show Context)
Citation Context ...ime argment, and we shall reprove it below (in fact, we shall give two proofs). This theorem was then generalized substantially by Szemerédi in 1975: Theorem 1.2 (Szemerédi’s theorem, first version). =-=[38]-=-, [39] Let A ⊂ Z + be a subset of 1 integers with positive upper density, thus lim supN→∞ |A ∩ [1, N]| > 0, and let k � 3. N Then A contains infinitely many arithmetic progressions n, n + r, . . .,n +... |

46 | Universal characteristic factors and Furstenberg averages
- Ziegler
(Show Context)
Citation Context ...s somewhat large; see [25] for a rigorous statement and proof of this “inverse theorem for the U 3 norm”. (Interestingly, there are some closely related results arising from ergodic theory; see [29], =-=[47]-=-). This concludes our discussion of Gowers’ proof of Szemerédi’s theorem for progressions of length 4; the argument also extends to higher k (see [17]) though with some non-trivial additional difficul... |

44 |
Representation of an odd number as a sum of three primes
- Vinogradov
- 1937
(Show Context)
Citation Context ...s the prime tuple conjecture would imply the strong Goldbach conjecture for sufficiently large N. The weak Goldbach conjecture, which is essentially proven (thanks primarily to the work of Vinogradov =-=[46]-=-), asserts that every odd number N larger than 5 can be written as the sum of three primes. (By “essentially proven” I mean that this conjecture has been verified for N � 10 17 and also rigourously pr... |

42 |
On certain sets of positive density
- Varnavides
- 1959
(Show Context)
Citation Context ...ession of length three in A. (In fact we have demonstrated � c ′ (3, δ)N 2 such progressions for some c ′ (3, δ) > 0). Proof. [Second version implies third version] This argument is due to Varnavides =-=[45]-=-. We first observe that to prove the theorem, it suffices to do so when f is a characteristic function f = 1A. This is because if f is non-negative, bounded and obeys (2.1) then the set A := {n ∈ Z/NZ... |

33 | Applications of the regularity lemma for uniform hypergraphs, submitted
- Rödl, Skokan
(Show Context)
Citation Context ...roofs; a proof of Gowers [16], [17] in 2001 which combines “higher order” Fourier analytic methods with techniques from additive combinatorics; and also arguments of Gowers [18] and Rodl-Skokan [34], =-=[35]-=- using the machinery of hypergraphs. While we will not discuss all these separate proofs in detail here, we will need to discuss certain ideas from each of these arguments as they will eventually be u... |

33 | A quantitative ergodic theory proof of Szemerédi’s theorem
- Tao
- 2006
(Show Context)
Citation Context ... theorem] We now give an energy increment proof of Roth’s theorem, inspired by arguments of Furstenberg [10], Bourgain [4], and Green [20], as well as later arguments by Green and the author in [24], =-=[41]-=-. This is not the shortest such proof, nor the most efficient as far as explicit bounds are concerned, but it is a proof which has a relatively small reliance on Fourier analysis and thus which genera... |

31 | Multidimensional van der Corput and sublevel set estimates
- Carbery, Christ, et al.
- 1999
(Show Context)
Citation Context ...‖□ 1 = |E(K(x)|x ∈ A1)| which is degenerate and thus not a genuine norm. The significance of the Gowers cube norm to expressions of the form (3.1) lies in the following estimate (which is implicit in =-=[8]-=- and also in [17]).ARITHMETIC PROGRESSIONS AND PRIMES 17 Lemma 3.1 (Van der Corput lemma). Let d � 1, let A1, . . .,Ad be finite non-empty sets, let K : ∏d j=1 Aj → C, and for each 1 � i � d let Fi :... |

28 | Higher correlations of divisor sums related to primes. I. Triple correlations
- Goldston, Yıldırım
(Show Context)
Citation Context ...1PR is a bit too “rough” to serve as a good weight function, and it is better to use a slightly “smoother” variant of this function, namely the truncated divisor sums studied by Goldston and Yildirim =-=[13]-=-, [14], [15]. These are formed by replacing the von Mangoldt function Λ(n) = ∑ with the variant d|n ΛR(n) := ∑ d|n µ(d) log n d µ(d)(log R d )+ where x+ := max(x, 0) is the positive part of x. One can... |

22 |
Quasi-random subsets of Zn
- Chung, Graham
- 1992
(Show Context)
Citation Context ... 2 norm). Thus one can view the □ d norm as a multilinear generalization of the l 4 Schatten-von Neumann norm. This norm has also arisen in the study of pseudorandom sets and graphs, see for instance =-=[6]-=-. Now we specialize to the problem of counting arithmetic progressions in Z/NZ. Definition 3.3 (Gowers uniformity norm). Let f : Z/NZ → C be a function and d � 1. Then we define the Gowers uniformity ... |

19 |
A note on a question of Erdős and
- Solymosi
(Show Context)
Citation Context ... based on density increment arguments and extremely large cubes (see [19]), and an argument based on the Szemerédi regularity lemma (which in turn requires energy increment arguments in the proof) in =-=[37]-=-. While these arguments are also important to the theory and both have generalizations to higher k, we will not discuss them here due to lack of space. 3. Interlude on multilinear operators We will sh... |

19 |
der Corput, Uber Summen von Primzahlen und Primzahlquadraten
- Van
- 1939
(Show Context)
Citation Context ...s they will eventually be used in the proof of Theorem 5.1 below. The above theorems do not apply directly to the set of prime numbers, as they have density zero. Nevertheless, in 1939 van der Corput =-=[43]-=- proved, by using Fourier analytic methods (the Hardy-Littlewood circle method) which were somewhat similar to the methods used by Roth, the following result: 1991 Mathematics Subject Classification. ... |

14 | Restriction theory of the Selberg sieve, with applications
- Green, Tao
(Show Context)
Citation Context ...h (7.4)) to show that the algorithm to find B halts after only a bounded number of iterations. We now briefly remark on the earlier k = 3 versions of the above argument, referring the reader to [20], =-=[24]-=- for further details. In that case, the notion of pseudorandomness of the dominating measure ν was replaced by that of linear pseudorandomness or Fourier pseudorandomness, which basically asserts that... |

12 | A mean ergodic theorem for (1/N) ∑N n=1 f(T nx)g(T n2x). Convergence in ergodic theory and probability 92 - Weiss - 1993 |

11 | Linear relations amongst sums of two squares, Number Theory and Algebraic - Heath-Brown - 2003 |

9 |
Linear equations in primes, Mathematika 39
- Balog
- 1992
(Show Context)
Citation Context ...ese multilinear averages (the “rank one” averages involving three or more copies of Λ) can be treated by Fourier methods; this includes Vinogradov’s theorem and van der Corput’s theorem, and see also =-=[2]-=- for further discussion. However, it is by now well established that these techniques cannot directly extend to handle other multilinear averages. The k = 4 result in Theorem 5.1 requires a “quadratic... |

9 | Additive properties of dense subsets of sifted sequences - Ramaré, Ruzsa |

8 |
Pointwise convergence of ergodic averages along cubes
- Assani
(Show Context)
Citation Context ...th let us call a function f : A → C on a finite set A bounded if |f(n)| � 1 for all n ∈ A. Theorem 2.2 (Roth’s theorem, third version). Let 0 < δ � 1, and let N � 1 be a prime integer. Let f : Z/NZ → =-=[0, 1]-=- be a non-negative bounded function with large mean E(f(n)|n ∈ Z/NZ) � δ. (2.1) Then we have n∈A E(f(n)f(n + r)f(n + 2r)|n, r ∈ Z/NZ) � c(3, δ) − oδ(1) (2.2) for some c(3, δ) > 0 depending only on δ, ... |

8 | Three primes and an almost prime in arithmetic progression - Heath-Brown - 1981 |

8 | Twenty-two primes in arithmetic progression - Pritchard, Moran, et al. - 1995 |

7 |
There exists an infinity of 3-combinations of primes
- Chowla
- 1944
(Show Context)
Citation Context ...hall explain later, the k = 2 case is much more difficult and well beyond the reach of existing techniques. Now we turn to arithmetic progressions in the primes. In 1933 van der Corput [43] (see also =-=[7]-=-) established that the primes contain infinitely many arithmetic progressions of length 3; indeed we know the significantly stronger statement that the Hardy-Littlewood conjecture holds in this case, ... |

7 | On Snirel’man’s constant, Ann - Ramaré - 1995 |

6 |
Roth’s theorem for the primes, preprint
- Green
(Show Context)
Citation Context ...ength three, but also to obtain an asymptotic count as to how many such progressions there are; we shall return to this point later. Roth’s theorem and van der Corput’s theorem were combined by Green =-=[20]-=- in 2003 to obtain Theorem 1.4 (Green’s theorem). [20] Let A ⊂ P be a subset of primes with positive relative upper density: |A ∩ [1, N]| lim sup > 0. N→∞ |P ∩ [1, N]| Then A contains infinitely many ... |

5 |
A Szemeredi-type theorem for sets of positive density
- Bourgain
(Show Context)
Citation Context ...trivial arithmetic of length three in A. Proof. [Energy increment proof of Roth’s theorem] We now give an energy increment proof of Roth’s theorem, inspired by arguments of Furstenberg [10], Bourgain =-=[4]-=-, and Green [20], as well as later arguments by Green and the author in [24], [41]. This is not the shortest such proof, nor the most efficient as far as explicit bounds are concerned, but it is a pro... |

5 |
The ergodic-theoretical proof of Szemerédi’s theorem
- Furstenberg, Katznelson, et al.
- 1982
(Show Context)
Citation Context ...ry) and very complicated. A substantially shorter proof - but one involving the full machinery of measure theory and ergodic theory, as well as the axiom of choice - was obtained by Furstenberg [10], =-=[11]-=- in 1977. Since then, there have been two other types of proofs; a proof of Gowers [16], [17] in 2001 which combines “higher order” Fourier analytic methods with techniques from additive combinatorics... |

5 |
An inverse theorem for the Gowers U3 norm, preprint
- Green, Tao
(Show Context)
Citation Context ...ximately equal to a higher-dimensional linear function to again deduce a density increment of A on some sub-progression. This is again done mainly by Weyl’s theory of uniform distribution; however in =-=[25]-=- an alternate argument was developed, which is based on locating a primitive F to a. This argument closely mimics the one given in the one-dimensional case when a(h) ≈ αh + β; however, there is an add... |

4 |
gaps between primes
- Small
(Show Context)
Citation Context ... too “rough” to serve as a good weight function, and it is better to use a slightly “smoother” variant of this function, namely the truncated divisor sums studied by Goldston and Yildirim [13], [14], =-=[15]-=-. These are formed by replacing the von Mangoldt function Λ(n) = ∑ with the variant d|n ΛR(n) := ∑ d|n µ(d) log n d µ(d)(log R d )+ where x+ := max(x, 0) is the positive part of x. One can easily veri... |

4 |
ergodic averages and nilmanifolds
- Nonconventional
(Show Context)
Citation Context ..., χ〉 is somewhat large; see [25] for a rigorous statement and proof of this “inverse theorem for the U 3 norm”. (Interestingly, there are some closely related results arising from ergodic theory; see =-=[29]-=-, [47]). This concludes our discussion of Gowers’ proof of Szemerédi’s theorem for progressions of length 4; the argument also extends to higher k (see [17]) though with some non-trivial additional di... |

3 |
correlations of divisor sums related to primes, III: k-correlations, preprint (available at AIM preprints
- Higher
(Show Context)
Citation Context ... a bit too “rough” to serve as a good weight function, and it is better to use a slightly “smoother” variant of this function, namely the truncated divisor sums studied by Goldston and Yildirim [13], =-=[14]-=-, [15]. These are formed by replacing the von Mangoldt function Λ(n) = ∑ with the variant d|n ΛR(n) := ∑ d|n µ(d) log n d µ(d)(log R d )+ where x+ := max(x, 0) is the positive part of x. One can easil... |