= Ruzsa modelling lemma
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= Ruzsa modeling lemma
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{synonym}
For a nonempty finite <set> $A\subseteq\mathbb Z$ with bounded <doubling constant> and a fixed <positive integer> $s$, the <Ruzsa modelling lemma> finds a subset of size at least $c_s|A|$ that is <Freiman s-isomorphic> to a dense subset of a finite <cyclic group>. The model order is at most $C_{K,s}|A|$ when $|A+A|\leq K|A|$. The <cyclic Freiman model of a small-doubling integer set> gives an explicit order-eight version.
The integer, or appropriate torsion-free, setting matters. If $A\subseteq\mathbb F_2^n$, then $x+x=y+y$ for every $x,y\in A$. A <Freiman 2-isomorphism> into a <cyclic group> would force $2\phi(x)=2\phi(y)$. All image differences would lie in its <subgroup> of elements killed by two, which has at most two elements. Thus no subset with more than two <points> can be <Freiman 2-isomorphic> to a cyclic model. General <abelian groups> need models retaining their torsion, rather than an unrestricted cyclic-model assertion.
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