Freiman's theorem in an arbitrary abelian group

By Ben Joseph Green, Imre Z. Ruzsa. 2005

204 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/739b77054ce4f61d7a2e41dda01e0898f459eaf3

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Citations in the corpus

Near optimal bounds in Freiman's theorem (2011, methodology): “The next ingredient of our approach is Chang’s Spectral Lemma [3] (see also [10]).”
Nilprogressions and groups with moderate growth (2015, unknown): “, vr ;L) of bounded rank such that S 2n1 ⊂ H0P ⊂ SOε(n1) [22].”
On the Bogolyubov–Ruzsa lemma (2010, methodology): “With this in mind the extension from Z to arbitrary abelian groups is due to Green and Ruzsa [GR07], and the first good bounds to Schoen [Sch10] for certain classes of groups.”
Growth in SL_3(Z/pZ) (2008, background): “row truly rapidly (|A·A·A| ≫ |A|1+δ). All we know is which sets grow very slowly (|A·A·A| ≪ (log|A|)1/3|A|, say); this is Freiman’s theorem, generalised to arbitrary abelian groups by Green and Ruzsa [GR]). We will not attempt to improve on this; we will do no more than set aside the abelian case wh”
Decompositions, approximate structure, transference, and the Hahn–Banach theorem (2008, methodology): “A generalization of Freiman’s theorem to subsets of an arbitrary Abelian group was proved by Green and Ruzsa [10].”
An inverse theorem for the Gowers U^3 norm (2005, methodology): “et progression P+Hto be the same as that of its component P. The need to generalize from generalized arithmetic progressions to coset progressions in the setting of a general group Gwas ﬁrst noted in [33]. 13 We now use standard facts from the geometry of numbers to show that every Bohr set contains”
The structure theory of set addition revisited (2012, methodology): “In [Ruz99] Ruzsa proved the result for groups of bounded exponent which is in some sense at the other end of the spectrum from torsion-free, and then Green and Ruzsa in [GR07] established the result above for arbitrary (Abelian) groups with another proof appearing a little later in [TV06, Theorem 5.”
The structure of approximate groups (2011, methodology): “Again, see [47]; see also [24] for an earlier result in this direction.”
COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM (2005, background): “m for some m. In fact by the machinery of Freiman isomorphisms one can reduce further, to the case when A is a subset of the integers. The case of groups with torsion has some further subtleties; see [12]. As stated, Proposition 1.3 does not quite imply Freiman’s theorem, since we have not bounded d”
Open problems in additive combinatorics (2007, unknown): “Green and Ruzsa [50] extended Freiman’s theorem to arbitrary abelian groups.”
A Probabilistic Technique for Finding Almost-Periods of Convolutions (2010, methodology): “Partial progress was made on this question by Luczak and Schoen [28], who also noted that work of Green and Ruzsa [21] implies that one can always find an x ∈ A+A with 1A ∗ 1A(x) 6 max(1, |A|/(log2 p)1/2+o(1)).”
On the singularity probability of random Bernoulli matrices (2005, background): “ ≥ 1−10−4 then x ∈ A. Now we are going to show that the set of those x such that |h(x)| ≥ 1 − 10−4 has cardinality Θ(2n−d±). This proof is trickier and we need to adapt an argument from [7] (see also [8]), which requires the full strength of (7.8). Let k be a large integer to be chosen later, and fo”
Sets with Small Sumset and Rectification (2004, methodology): “With a future application in mind [ 8 ] we will describe the results here for general finite abelian groups.”