Roth’s theorem on progressions revisited

By Jean Bourgain. 2008

131 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d864b3f9a54c53633918a8aa61ebe3281ce7464b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Tom Sanders
2. Olof Sisask
3. Ernie Croot
4. Terence Tao
5. Ben Joseph Green
6. Emmanuel Breuillard

In Electrical and Electronic Engineering

1. Ernie Croot
2. Olof Sisask
3. Tom Sanders
4. Terence Tao

In Geometry and Topology

1. Tom Sanders
2. Olof Sisask
3. Ernie Croot
4. Terence Tao
5. Emmanuel Breuillard
6. Ben Joseph Green

Citations in the corpus

• A Probabilistic Technique for Finding Almost-Periods of Convolutions (2010, background): “Subsequent Fourier-analytic arguments have demonstrated better bounds for r3: the best bound currently known is due to Bourgain, who in [5] established that r3(N) ≪ (log logN) (logN)2/3 N.”
• On Roth's theorem on progressions (2010, background): “ot address ourselves to a comprehensive history here. The previous best bound was of the above shape with the power of 3/4 instead of 1 which comes from a refinement [San10] of an argument of Bourgain [Bou08]. The methods of this paper, however, are largely unrelated to those refinements. There are tw”
• The structure theory of set addition revisited (2012, background): “(See, for example, [Cha02] or the appendix to [Bou08] for the torsion-free case, and [GT09b] for the bounded exponent case.”
• On certain other sets of integers (2010, methodology): “n of (3.5) and we find that for A ⊂ G of density α we have (3.10) T(1A,1A,1A) > exp(−Oe(τ −2k 1α−1 +k +τ2α−2)). Of course τ ∈ [Ω(α), √ α] so we can optimise with k = α−3/2 to get the main result in [Bou08]. 3.5. Relaxing the dissociation. To get the density increment from (3.8) we used the fact th”
• On the Bogolyubov–Ruzsa lemma (2010, methodology): “Following [Bou08] we use a slight generalization of the traditional notion of Bohr set, letting the width parameter vary according to the character.”
• Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. (2020, unknown): “These are known as Bohr sets, which were first used in this context by Bourgain [6].”