Good Freiman models from polynomial coset covers

Definitions and mathematical statement

Let $G$ be an abelian group, let $A\subset G$ be finite and nonempty, and suppose $|A+A|\leq K|A|$ for a real $K\geq1$. A Freiman isomorphism of order two from $A'\subseteq A$ into an abelian group $H$ is an injective map $\phi:A'\to H$ such that, for every $a,b,c,d\in A'$, \[ a+b=c+d\quad\Longleftrightarrow\quad \phi(a)+\phi(b)=\phi(c)+\phi(d). \] Thus $\phi$ is a bijection onto its image and preserves both additive quadruple relations and their failure; it need not extend to a group homomorphism.

The question asks whether there are absolute constants $C\geq1$ and $c>0$ such that every such $G,A,K$ has a nonempty subset $A'$ and a finite abelian group $H$ with \[ |A'|\geq cK^{-C}|A|,\qquad |H|\leq CK^C|A|, \] and a Freiman isomorphism of order two from $A'$ into $H$. The constants must be independent of the original group's torsion, rank, and size. The model is permitted to change both the ambient group and the retained subset, but only a polynomial loss in $K$ is allowed. The upper bound measures the entire model group, not merely the image of $A'$. This formulation makes explicit the absolute constants hidden in the source's $K^{O(1)}$ notation and isolates its question for arbitrary abelian groups.

Short English statement

Does every finite set of small doubling contain a polynomially large subset with a comparably small finite additive model?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000902 (AIM-COMBINATORICS-0027), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Additive combinatorics and its applications, item 2.3. Original workshop question underlying AIM-COMBINATORICS-0027. http://aimpl.org/addcombapp/2/.