Roth's theorem in the primes

By Ben Joseph Green. 2003

217 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6c662f8d940ca4b794cac93aef61cee3da45df3a

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Citations in the corpus

Finite configurations in sparse sets (2013, background): “Green [18] proved the case k = 3, then with Tao [19] proved the general case k ≥ 3.”
Arithmetic Progressions in Sets of Fractional Dimension (2007, methodology): “We rely on many of the ideas from [24], [12], and particularly [14], [30], such as the use of the trilinear form Λ in a Fourier representation and a decomposition of the measure μ into “random” and “periodic” parts.”
The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view (2005, methodology): “where (N) is the number of primes in{1,...,N}, then for every integer k 1, A contains an arithmetic progression of length k. For k = 3, this was proved by Green [ 14 ].”
Linear equations in primes (2006, methodology): “This was a vital trick in [20; 24; 22], where it was used in similar fashion to eliminate the local factors p̌.”
On Roth's theorem on progressions (2010, methodology): “Indeed, while it seems likely that with care one could improve the 5 a bit, if one could replace it by 1−Ω(1), then one could use the W -trick (as popularised by Green [Gre05]) to deduce van der Corput’s theorem pretty easily; if one could replace it by −Ω(1), then van der Corput’s theorem would fol”
A transference approach to a Roth-type theorem in the squares (2015, methodology): “We overcome this difficulty by implementing the W -trick, as found in Green [Gre05].”
An inverse theorem for the Gowers U^3 norm (2005, methodology): “nd in particular has small U2 norm. The ﬁrst term can be written as a 1-step nilfunction, the underlying nilmanifold being (R/Z)d and the rotation Tbeing (x 1,...,xd) → (x 1 +r 1/N,...,xd +rd/N). See [29] for an application of such a decomposition (there the language of nilsystems and ergodic theory”
The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, methodology): “application of Vinogradov’s method, but can also be achieved using harmonic analysis methods arising from restriction theory; see [22], [28].”
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. (2020, unknown): “For example, we immediately obtain a strong form of a theorem originally due to Green [13]: every subset of the primes of positive relative density contains infinitely many non-trivial three-term arithmetic progressions.”