On the Bogolyubov–Ruzsa lemma

@article{Sanders2012OnTB,
  title={On the Bogolyubov–Ruzsa lemma},
  author={Tom Sanders},
  journal={Analysis \& PDE},
  year={2012},
  volume={5},
  pages={627-655}
}
  • T. Sanders
  • Published 30 October 2010
  • Mathematics
  • Analysis & PDE
Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|. 

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References

SHOWING 1-10 OF 68 REFERENCES

Freiman's theorem in an arbitrary abelian group

A famous result of Freiman describes the structure of finite sets A ⊆ ℤ with small doubling property. If |A + A| ⩽ K|A|, then A is contained within a multidimensional arithmetic progression of

On problems of Erdös and Rudin

Product set estimates for non-commutative groups

TLDR
A Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre is developed.

Roth’s theorem on progressions revisited

TLDR
An improvement of the density condition for a subset A to contain a nontrivial arithmetic progression of length 3 to prove the following Theorem.

Sets with Small Sumset and Rectification

We study the extent to which sets A ⊆ Z/NZ, N prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely

MULTIPLE SET ADDITION IN Z p

It is shown that there exists an absolute constant H such that for every h > H, every prime p, and every set A ⊆ Zp such that 10 ≤ |A| ≤ p(lnh)/(9h) and |hA| ≤ h3/2|A|/(8(lnh)1/2), the set A is

Equivalence of polynomial conjectures in additive combinatorics

TLDR
The main result is that the two conjectures in additive combinatorics, the polynomial Freiman-Ruzsa conjecture and the inverse Gowers conjecture for U3, are in fact equivalent.

A new proof of Szemerédi's theorem

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these

Additive combinatorics

  • T. TaoV. Vu
  • Mathematics
    Cambridge studies in advanced mathematics
  • 2007
TLDR
The circle method is introduced, which is a nice application of Fourier analytic techniques to additive problems and its other applications: Vinogradov without GRH, partitions, Waring’s problem.

Spectral Structure of Sets of Integers

Let E be a small subset of a finite abelian group, and let R be the set of points at which its Fourier transform is large. A result of Chang states that R has a great deal of additive structure. We
...