# 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} }

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

### The polynomial Freiman-Ruzsa conjecture

- Mathematics
- 2014

Original proposers of the open problem: Imre Ruzsa, Katalin Marton The year when the open problem was proposed: 1999 Sponsor of the submission: Timothy Gowers - University of Cambridge AMS Subject…

### Bilinear Bogolyubov Argument in Abelian Groups

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- 2021

and carry out sufficiently many steps where we replace every row or every column of A by the set difference of it with itself, then inside the resulting set we obtain a bilinear variety of…

### Small Doubling in Groups

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- 2013

Let A be a subset of a group G = (G; ·). We will survey the theory of sets A with the property that |A · A|≤K|A|, where A · A = {a1a2: a1; a2 ∈ A}. The case G = (ℤ; +) is the famous Freiman-Ruzsa…

### Freiman's theorem in an arbitrary nilpotent group

- Mathematics
- 2012

We prove a Freiman–Ruzsa‐type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K ‐approximate group A in an s ‐step nilpotent group G is contained in a coset nilprogression…

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- 2020

A qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularityLemma for sets of bounded VC-dimension in finite groups of bounded exponent are obtained.

### A model-theoretic note on the Freiman–Ruzsa theorem

- MathematicsSelecta Mathematica
- 2021

A non-quantitative version of the Freiman-Ruzsa theorem is obtained for finite stable sets
with small tripling in arbitrary groups, as well as for (finite) weakly normal subsets in
abelian groups.

### An Exposition of Sanders' Quasi-Polynomial Freiman-Ruzsa Theorem

- MathematicsTheory Comput.
- 2012

The aim of this note is to make the proof of the polynomial Freiman-Ruzsa conjecture accessible to the theoretical computer science community, and in particular to readers who are less familiar with additive combinatorics.

### Quantitative structure of stable sets in finite abelian groups

- MathematicsTransactions of the American Mathematical Society
- 2020

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A…

### Sums of Linear Transformations in Higher Dimensions

- MathematicsThe Quarterly Journal of Mathematics
- 2019

In this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 10 depending only on q, s and d such that for any finite subset A of ℝd that is…

### Difference sets are not multiplicatively closed

- Mathematics
- 2016

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar…

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