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. Green, I. 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. 
Near optimal bounds in Freiman's theorem
We prove that if for a finite set A of integers we have |A+A| ≤ K|A|, then A is contained in a generalized arithmetic progression of dimension at most K −1/2 and size at most exp(K −1/2 )|A| for some
A Freiman-type theorem for locally compact abelian groups
We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) d log d then we describe A in a way which is
An application of a local version of Chang's theorem
We prove a theorem claimed in math.CA/0605519 which asserts that if A is a subset of a compact abelian group G with density of a particular (natural, although technical) form then the A(G)-norm (that
On a theorem of Shkredov
We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\log |A|) such that |A \cap Span(L)| >> c^{1/3}|A|.
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
Approximate groups and doubling metrics
  • T. Sanders
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2011
Abstract We develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small
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
Mathematical Proceedings of the Cambridge Philosophical Society
We develop a version of Freı̆man’s theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling
EXTENSIONS OF SCHREIBER’S THEOREM ON DISCRETE APPROXIMATE SUBGROUPS IN R d
. — In this paper we give an alternative proof of Schreiber’s theorem which says that an infinite discrete approximate subgroup in R d is relatively dense around a subspace. We also deduce from
...
...

References

SHOWING 1-10 OF 34 REFERENCES
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
TLDR
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
Sur quelques propriétés arithmétiques des presque-périodes
  • Ann. Chaire Math. Phys. Kiev
  • 1939
...
...