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The primes contain arbitrarily long arithmetic progressions

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of… Expand

Linear equations in primes

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation… Expand

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

- B. Green
- Mathematics
- 30 October 2003

Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the… Expand

Roth's theorem in the primes

- B. Green
- Mathematics
- 25 February 2003

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… Expand

The quantitative behaviour of polynomial orbits on nilmanifolds

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper… Expand

The structure of approximate groups

- E. Breuillard, B. Green, T. Tao
- Mathematics
- 22 October 2011

Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A⋅A is covered by K left translates of A.The main result of… Expand

Approximate Subgroups of Linear Groups

- E. Breuillard, B. Green, T. Tao
- Mathematics
- 11 May 2010

We establish various results on the structure of approximate subgroups in linear groups such as SLn(k) that were previously announced by the authors. For example, generalising a result of Helfgott… Expand

On Sets Defining Few Ordinary Lines

TLDR

Sum-free sets in abelian groups

LetA be a subset of an abelian groupG with |G|=n. We say thatA is sum-free if there do not existx, y, z εA withx+y=z. We determine, for anyG, the maximal densityμ(G) of a sum-free subset ofG. This… Expand

The Mobius function is strongly orthogonal to nilsequences

We show that the Möbius function μ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F (g(n)Γ))n∈N. Here, G is a simply-connected nilpotent Lie group with a discrete and… Expand

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