On the Bogolyubov–Ruzsa lemma

  title={On the Bogolyubov–Ruzsa lemma},
  author={Tom Sanders},
  journal={Analysis \& PDE},
  • 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|. 

The polynomial Freiman-Ruzsa conjecture

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

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

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

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

On finite sets of small tripling or small alternation in arbitrary groups

  • G. Conant
  • Mathematics
    Combinatorics, Probability and Computing
  • 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

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

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

  • C. TerryJ. Wolf
  • Mathematics
    Transactions 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

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

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



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

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

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


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

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
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