Freiman's theorem in an arbitrary abelian group

@article{Green2005FreimansTI,
  title={Freiman's theorem in an arbitrary abelian group},
  author={Ben Green and Imre Z. Ruzsa},
  journal={Journal of the London Mathematical Society},
  year={2005},
  volume={75}
}
  • B. GreenI. Ruzsa
  • Published 10 May 2005
  • Mathematics
  • Journal of the London Mathematical Society
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 dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group. 

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References

SHOWING 1-10 OF 34 REFERENCES

An analog of Freiman's theorem in groups

It is proved that any set A in a commutative group G where the order of elements is bounded by an integer r having n elements and at most n sums is contained in a subgroup of size An with A = f(r; )

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

Fourier Analysis on Groups

In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact

A polynomial bound in Freiman's theorem

.Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s

On Small Sumsets in (ℤ/2ℤ)n

It is proved that any subset of (ℤ/2 ℤ)n, having k elements, such that c = c means c is contained in a subgroup of order at most u−1k where u=u(c)>0 is an explicit function of c which does not depend on k nor on n.

An Introduction to the Geometry of Numbers

Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction

Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics 12 (WileyInterscience

  • New York,
  • 1962

Generalized arithmetical progressions and sumsets

Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12 Interscience Publishers (a division of John Wiley and Sons)

  • 1962