Benjamin Green

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Author Description

Benjamin (Ben) Green is a British mathematician known for his work in additive combinatorics and number theory. He is a professor at the University of Oxford, holding the Waynflete Professorship of Pure Mathematics. One of his most famous results, in collaboration with Terence Tao, is the Green–Tao theorem, proving that there are arbitrarily long arithmetic progressions of prime numbers.

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All published works (1)

The primes contain arbitrarily long arithmetic progressions

2008-03-01

Benjamin Green , Terence Tao

We prove that there are arbitrarily long arithmetic progressions of primes.There are three major ingredients.The first is Szemerédi's theorem, which asserts that any subset of the integers of positive density … We prove that there are arbitrarily long arithmetic progressions of primes.There are three major ingredients.The first is Szemerédi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length.The second, which is the main new ingredient of this paper, is a certain transference principle.This allows us to deduce from Szemerédi's theorem that any subset of a sufficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length.The third ingredient is a recent result of Goldston and Yıldırım, which we reproduce here.Using this, one may place (a large fraction of) the primes inside a pseudorandom set of "almost primes" (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density.

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Common Coauthors

Coauthor	Papers Together
Terence Tao	1

Commonly Cited References

ON THE REPRESENTATION OF A LARGER EVEN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF AT MOST TWO PRIMES

1973-02-20

Chen Jing-Run

In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is … In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is simple without any complicated numerical calculations.

Higher correlations of divisor sums related to primes III: k-correlations

2002-01-01

D. A. Goldston , C. Y. Yıldırım

We obtain the general k-correlations for a short divisor sum related to primes. We obtain the general k-correlations for a short divisor sum related to primes.

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Polynomial extensions of van der Waerden’s and Szemerédi’s theorems

1996-01-01

Vitaly Bergelson , A. Leibman

An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: … An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S subset-of-or-equal-to double-struck upper Z Superscript l"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>l</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">S\subseteq \mathbb {Z}^l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a set of positive upper Banach density, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p 1 left-parenthesis n right-parenthesis comma ellipsis comma p Subscript k Baseline left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p_1(n),\dotsc ,p_k(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be polynomials with rational coefficients taking integer values on the integers and satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript i Baseline left-parenthesis 0 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_i(0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i equals 1 comma ellipsis comma k semicolon"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>;</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">i=1,\dotsc ,k;</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v 1 comma ellipsis comma v Subscript k Baseline element-of double-struck upper Z Superscript l"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>l</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">v_1,\dotsc ,v_k\in \mathbb {Z}^l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exist an integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a vector <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u element-of double-struck upper Z Superscript l"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>l</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">u\in \mathbb {Z}^l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u plus p Subscript i Baseline left-parenthesis n right-parenthesis v Subscript i element-of upper S"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">u+p_i(n)v_i\in S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i less-than-or-equal-to k"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i\le k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Arithmetic progressions of length three in subsets of a random set

1996-01-01

Yoshiharu Kohayakawa , Tomasz Łuczak , Vojtěch Rödl

Arithmetic progressions of length three in subsets of Arithmetic progressions of length three in subsets of

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Small Gaps Between Primes I

2005-01-01

D. A. Goldston , C. Y. Yıldırım

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that … We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes.

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On sets of integers not containing long arithmetic progressions

2001-01-01

Izabella Łaba , Michael T. Lacey

We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets … We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic progressions of length 3.

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A Szemeredi-type regularity lemma in abelian groups

2003-10-30

Ben Green

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Pointwise convergence of ergodic averages along cubes

2010-01-01

Idris Assani

Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions

1977-12-01

Harry Furstenberg

On Triples in Arithmetic Progression

1999-12-01

Jean Bourgain

A Quantitative Ergodic Theory Proof of Szemerédi's Theorem

2006-11-06

Terence Tao

A famous theorem of Szemerédi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain … A famous theorem of Szemerédi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemerédi's original combinatorial proof using the Szemerédi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and the more recent proofs of Gowers and Rödl-Skokan using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring passage (via the Furstenberg correspondence principle) to an infinitary measure preserving system, and then decomposing a general ergodic system relative to a tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is "elementary" in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

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Three Primes and an Almost-Prime in Arithmetic Progression

1981-06-01

I. Heath-Brown

A Szemeredi-type regularity lemma in abelian groups, with applications

2003-01-01

Ben Green

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of … Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorm for sets which are almost sum-free. If A is a subset of [N] which contains just o(N^2) triples (x,y,z) such that x + y = z then A may be written as the union of B and C, where B is sum-free and |C| = o(N). Another answers a question of Bergelson, Host and Kra. If alpha, epsilon > 0, if N > N_0(alpha,epsilon) and if A is a subset of {1,...,N} of size alpha N, then there is some non-zero d such that A contains at least (alpha^3 - epsilon)N three-term arithmetic progressions with common difference d.

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Restriction theory of the Selberg sieve, with applications

2006-01-01

Ben Green , Terence Tao

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 … 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 a 1 ,⋯,a k and b 1 ,⋯,b k be positive integers. Write h(θ):=∑ n∈X e(nθ), where X is the set of all n≤N such that the numbers a 1 n+b 1 ,⋯,a k n+b k are all prime. We obtain upper bounds for ∥h∥ L p (𝕋) , p>2, which are (conditionally on the Hardy-Littlewood 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 p 1 <p 2 <p 3 of primes, such that p i +2 is either a prime or a product of two primes for each i=1,2,3.

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Higher correlations of divisor sums related to primes III: small gaps between primes

2007-07-16

D. A. Goldston , C. Y. Yıldırım

We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for … We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η > 0, a positive proportion of consecutive primes are within 4 + η times the average spacing between primes. Authors’ note. This paper was written in 2004, prior to the solution, in [8], of the problem considered here. In [8] it is shown that Δ = 0. While the main result in Theorem 1 has now been superseded, we believe the method used here is both of interest and future utility in other applications. In particular, the work of Green and Tao [12] on arithmetic progressions of primes makes use of Proposition 1 of this paper.

Twenty-Two Primes in Arithmetic Progression

1995-07-01

Paul Pritchard , Andrew Moran , Anthony Thyssen

Some newly-discovered arithmetic progressions of primes are presented, including five of length twenty-one and one of length twenty-two. Some newly-discovered arithmetic progressions of primes are presented, including five of length twenty-one and one of length twenty-two.

Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes

1923-01-01

G. H. Hardy , J. E. Littlewood

z.I.It was asserted by GOLDBACH, in a letter to "EuLER dated 7 June, 1742 , that every even number 2m is the sum o/two odd primes, ai~d this propos ition … z.I.It was asserted by GOLDBACH, in a letter to "EuLER dated 7 June, 1742 , that every even number 2m is the sum o/two odd primes, ai~d this propos ition has generally been described as 'Goldbach's Theorem'.There is no reasonable doubt that the theorem is correct, and that the number of representations is large when m is large; but all attempts to obtain a proof have been completely unsuccessful.Indeed it has never been shown that every number (or every large number, any number, that is to say, from a certain point onwards) is the sum of xo primes, or of i oooooo; and the problem was quite recently classified as among those 'beim gegenwiirtigen Stande der Wissensehaft unangreifbar'.~In this memoir we attack the problem with the aid of our new transcendental method in 'additiver Zahlentheorie'.~ We do not solve it: we do not

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On Certain Sets of Positive Density

1959-07-01

P. Varnavides

Roth’s theorem in the primes

2005-05-01

Ben Green

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 … 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.

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A szemerédi type theorem for sets of positive density inR k

1986-10-01

Jean Bourgain

Convergence of polynomial ergodic averages

2005-12-01

Bernard Host , Bryna Kra

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An ergodic Szemerédi theorem for commuting transformations

1978-12-01

Hillel Fürstenberg , Yitzhak Katznelson

Über Summen von Primzahlen und Primzahlquadraten

1939-12-01

J. G. van der Corput

Linear equations in primes

1992-12-01

Antal Balog

Six Primes and an Almost Prime in Four Linear Equations

1998-06-01

Antal Balog

Abstract There are infinitely many triplets of primes p, q, r such that the arithmetic means of any two of them, are also primes. We give an asymptotic formula for … Abstract There are infinitely many triplets of primes p, q, r such that the arithmetic means of any two of them, are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.

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A density version of the Hales-Jewett theorem

1991-12-01

Hillel Fürstenberg , Yitzhak Katznelson

On Certain Sets of Integers

1953-01-01

K. F. Roth

Convergence of Conze–Lesigne averages

2001-03-30

Bernard Host , Bryna Kra

We study the convergence of N ^{-1} \sum f _1( T ^{a_1n} x ) f _2( T ^{a_2n} x ) f _3( T ^{a_3n} x ), for a measure-preserving system … We study the convergence of N ^{-1} \sum f _1( T ^{a_1n} x ) f _2( T ^{a_2n} x ) f _3( T ^{a_3n} x ), for a measure-preserving system ( X , \mathcal{B}, \mu, T ) and f _{1}, f _{2}, f _{3} \in L ^{\infty}(\mu). This generalizes the theorem of Conze and Lesigne on such expressions and simplifies the proof. We also obtain a description of the limit.

On sets of integers containing no four elements in arithmetic progression

1969-03-01

Endre Szemerédi

Nonconventional ergodic averages and nilmanifolds

2005-01-01

Bernard Host , Bryna Kra

We study the L 2 -convergence of two types of ergodic averages.The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as … We study the L 2 -convergence of two types of ergodic averages.The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg's proof of Szemerédi's theorem.The second average is taken along cubes whose sizes tend to +∞.For each average, we show that it is sufficient to prove the convergence for special systems, the characteristic factors.We build these factors in a general way, independent of the type of the average.To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold.From the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density.* .By Proposition 3.4, this set depends only on the first coordinate.This means that there exists a subset B of X with X × A = B × X [k] * , up to a subset of X [k] of µ [k] -measure zero.That is, 1 A (x) = 1 B (x 0 ) for µ [k] -almost every x = (x 0 , x) ∈ X [k] .(12)

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The ergodic theoretical proof of Szemerédi’s theorem

1982-01-01

Hillel Fürstenberg , Yitzhak Katznelson , Donald Ornstein

Introduction. In Introduction. In

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XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression

1961-01-01

R. A. Rankin

SYNOPSIS Sets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or … SYNOPSIS Sets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have been obtained only for n =3. The problem is generalized in various ways. The analysis can also be applied to construct sets for the analogous problem of geometrical progressions. These sets are of positive density, unlike those of the first kind, which have zero density.

Additive properties of dense subsets of sifted sequences

2001-01-01

Olivier Ramaré , Imre Z. Ruzsa

We examine additive properties of dense subsets of sifted sequences, and in particular prove under very general assumptions that such a sequence is an additive asymptotic basis whose order is … We examine additive properties of dense subsets of sifted sequences, and in particular prove under very general assumptions that such a sequence is an additive asymptotic basis whose order is very well controlled.

A New Proof of Szemer�di's Theorem for Arithmetic Progressions of Length Four

1998-07-01

W. T. Gowers

Universal characteristic factors and Furstenberg averages

2006-03-17

Tamar Ziegler

Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and … Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and a_j are integers. A factor of X is characteristic for averaging schemes of length k (or k-characteristic) if for any non zero distinct integers a_1,...,a_k, the limiting L^2(\mu) behavior of the averages in (*) is unaltered if we first project the functions f_j onto the factor. A factor of X is a k-universal characteristic factor (k-u.c.f)} if it is a k-characteristic factor, and a factor of any k-characteristic factor. We show that there exists a unique k-u.c.f, and it has a structure of a (k-1)-step nilsystem, more specifically an inverse limit of (k-1)-step nilflows. Using this we show that the averages in (*) converge in L^2(\mu). This provides an alternative proof to the one given by Host and Kra in 2002.

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A non-conventional ergodic theorem for a nilsystem

2005-05-19

Tamar Ziegler

We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit. We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.

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Twenty-two primes in arithmetic progression

1995-01-01

Paul Pritchard , Andrew E. Moran , Anthony Thyssen

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Higher Correlations of Divisor Sums Related to Primes I: Triple Correlations

2003-04-22

D. A. Goldston , C.Y. Yildirim

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