Foundations of a Structural Theory of Set Addition

By G. Freiman. 2007

411 citations recorded by the source corpus.

Subfields: Industrial and Manufacturing Engineering

View Paper: https://www.semanticscholar.org/paper/5892089a57951241489ac133f0b9bcd636d848e1

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In Industrial and Manufacturing Engineering

Citations in the corpus

An analog of Freiman's theorem in groups (2018, neutral): “A famous theorem of Freiman [1, 2] provides a certain structural description of these sets; in one of the possible formulations, it says that A can be covered by a generalized arithmetic progression where d < c and U < Cn with C depending on c.”
Quantitative stability for sumsets in $R^n$ (2014, background): “We can now prove the following important result, which is just a special case of Freiman’s “3k − 3 Theorem” [8, 9].”
Long arithmetic progressions in sumsets: Thresholds and bounds (2005, background): “The following is a consequence of another theorem of Freiman [12].”
Approximate lattices (2016, background): “By definition, every k-Freiman homomorphism extends uniquely to a map ϕ̂ : Λk → Ξk satisfying ϕ̂(gh) = ϕ̂(g)ϕ̂(h) for all g ∈ Λl, h ∈ Λm as long as l+m ≤ k.”
Freiman's theorem in an arbitrary abelian group (2005, background): “If |A+A| = K|A| then we say that A has doubling K, and if K is very small in comparison to |A| (we will be deliberately vague about what we mean by this) then we say that A has small doubling.”
On a question of Erdős and Moser (2005, methodology): “A fundamental result in additive number theory, proved by Freiman [7], tells us that this is basically the only example of sets with small sumsets.”
A Probabilistic Technique for Finding Almost-Periods of Convolutions (2010, methodology): “In the abelian setting the non-local version of this result is in the same vein as a result of Freiman, Halberstam and Ruzsa [17] that finds long arithmetic progressions or Bohr sets in A + A + A (see also [44, Theorem 4.43]); the best bounds currently known in this direction are due to Sanders [36]”
COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM (2005, background): “In this direction, we have a remarkable result of Freiman and Bilu [1, 2, 6], which shows that given any finite subset A of a torsion-free abelian group with σ [A] = K and given any ε > 0, one can find a progression P ′ with dimension at most log2 K + ε and size at most |A| such that |A ∩ P ′| exp(−”
The structure of approximate groups (2011, neutral): “The concept of a local homomorphism is closely related to that of a Freiman homomorphism in additive combinatorics, as discussed for example in [53].”
Small Ball Probability, Inverse Theorems, and Applications (2012, background): “In the 1970s, Freiman considered the collection of pairwise sums A+A := {a+a′|a, a′ ∈ A} [15].”
Groups, Orders, and Dynamics (2014, neutral): “t AB= 0 is an easy exercise.) 1.4.4 More Combinatorial Properties Recently, orderable groups have been considered as a natural framework to extend certain basic results of Additive Combinatorics (see [67, 132, 167] as general references). One of the most elementary ones is the inequality for product”