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. 
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161 Citations
Highly Influential Citations
41
Background Citations
63
Methods Citations
41
Results Citations
8

161 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
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
TLDR
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
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
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
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
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.
. 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…
Partition regularity and the primes
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Let $\Lambda\subseteq \{1,\ldots, N\}$ and let $\{a_{n}\}_{n\in\Lambda}$ be a sequence with $|a_n|\leq 1$ for all $n$. It is easy to see that\[…
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