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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 ofExpand
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 generalisationExpand
A Szemerédi-type regularity lemma in abelian groups, with applications
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 theExpand
Roth's theorem in the primes
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-LittlewoodExpand
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 paperExpand
The structure of approximate groups
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 ofExpand
Approximate Subgroups of Linear Groups
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 HelfgottExpand
On Sets Defining Few Ordinary Lines
  • B. Green, T. Tao
  • Mathematics, Computer Science
  • Discret. Comput. Geom.
  • 23 August 2012
Let $$P$$P be a set of $$n$$n points in the plane, not all on a line. Expand
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. ThisExpand
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 andExpand