# 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|.
141 Citations

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• Mathematics
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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

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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

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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.

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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