# 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}
}
• B. Green
• Published 25 February 2003
• Mathematics
• Annals of Mathematics
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

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

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

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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

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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 $M$.

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

. 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