# Roth's theorem in the primes

@article{Green2003RothsTI, title={Roth's theorem in the primes}, author={Ben Green}, journal={Annals of Mathematics}, year={2003}, volume={161}, pages={1609-1636} }

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.

## 163 Citations

### Ju l 2 00 3 Roth ’ s Theorem in the Primes

- Mathematics
- 2008

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…

### 2 5 Fe b 20 03 Roth ’ s Theorem in the Primes

- Mathematics
- 2008

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 socalled Hardy-Littlewood…

### Szemerédi's Theorem in the Primes

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2018

Abstract Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any…

### On sums of sets of primes with positive relative density

- MathematicsJ. Lond. Math. Soc.
- 2011

The argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.

### A Density Version of the Vinogradov Three Primes Theorem

- Mathematics
- 2012

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes…

### The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view

- Mathematics
- 2005

A long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput,…

### A New Proof of Vinogradov's Three Primes Theorem

- Mathematics
- 2013

We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory…

### Arithmetic structures in random sets

- Mathematics
- 2007

We extend two well-known results in additive number theory, Sarkozy’s theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of…

### (2019). Szemerédi's theorem in the primes. Proceedings of the Edinburgh Mathematical Society , 62 (2), 443-457.

- Mathematics
- 2018

. Green and Tao famously proved in 2005 that any subset of the primes of ﬁxed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of…

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