 Ruzsa modelling lemma

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 $|A+A|\le K|A|$. 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 $2\varphi (x)=2\varphi (y)$. 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.