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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
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
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 the
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-Littlewood
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 of
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
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 Helfgott
Freiman's theorem in an arbitrary abelian group
A famous result of Freiman describes the structure of finite sets A ⊆ ℤ with small doubling property. If |A + A| ⩽ K|A|, then A is contained within a multidimensional arithmetic progression of
On Sets Defining Few Ordinary Lines
It is shown that if large P is large then there are at least at least two lines passing through exactly two points of P, which confirms, for large P, a conjecture of Dirac and Motzkin, and solves the “orchard-planting problem”.
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