# 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.
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
• Mathematics
Proceedings 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
• Mathematics
J. 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
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
• 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.
. 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

## References

SHOWING 1-10 OF 49 REFERENCES
A restriction theorem for the Fourier transform
. In this note we will prove a (L , LP) -restriction theorem for certain submanifolds & of codimension / > 1 in an n— dimensional Euclidean space which arise as orbits under the action of a compact
Counting sumsets and sum-free sets modulo a prime
• Mathematics, Computer Science
• 2004
The number of distinct sets of the form A of residues modulo p that are said to be sum-free if there are no solutions to a = a′ + a″ with a, a′, a″ ∈ A is counted.
Exponential sums over primes in an arithmetic progression
• Mathematics
• 1985
In 1979 A. F. Lavrik obtained some estimates for exponential sums over primes in arithmetic progressions by an analytic method. In the present paper we give an estimate for the same sums, comparable
Integer Sets Containing No Arithmetic Progressions
lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of
An Introduction to the Theory of Numbers
• E. T.
• Mathematics
Nature
• 1946
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.
On Λ(p)-subsets of squares
This paper is a follow up of [B1]. It is shown that the sequence of squares {n2|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic
Multiplicative Number Theory
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The
Restriction and Kakeya phenomena for finite fields
• Mathematics
• 2002
The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this
On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.
• F. Behrend
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 1946
By a modification of Salem and Spencer' method, the better estimate 1-_2/2log2 + e v(N) > N VloggN is shown.
On the Hardy–Littlewood majorant problem
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2004
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\[