Compact Freiman models and relation peeling

Definitions and mathematical statement

Let $A,B$ be finite subsets of $\mathbb Z$. A Freiman isomorphism of order two is a bijection $\phi:A\to B$ such that, for every $a_1,a_2,a_3,a_4\in A$, allowing repetitions, \[ a_1+a_2=a_3+a_4 \quad\Longleftrightarrow\quad \phi(a_1)+\phi(a_2)=\phi(a_3)+\phi(a_4). \] Thus it preserves exactly all additive relations involving two summands on each side, in both directions. The length of a nonempty integer set $B$ means its diameter $\max B-\min B$, not its cardinality or the number of integers in its containing interval. Write $\ell(A)$ for the minimum length of an integer set Freiman isomorphic to $A$.

The Konyagin--Lev conjecture in the workshop is that for every integer $n\ge7$ and every $n$-element set $A\subseteq\mathbb Z$, \[ \ell(A)\le2^{n-2}. \] Equivalently, there must be such a bijection from $A$ to a subset of the integer interval $[0,2^{n-2}]$; translation puts a minimal model's smallest element at zero. The restriction $n\ge7$ is part of the conjecture. The exponential quantity arises from the model $\{0,1,2,4,\ldots,2^{n-2}\}$, whose repeated doubling relations motivate the proposed extremal length. A Freiman isomorphism is not required to be the restriction of an affine map of $\mathbb Z$; that extra requirement would change the problem. The exponent in the interval endpoint is restored from the original PDF typography.

Short English statement

Can every n-element integer set, for n at least seven, be modeled with exactly the same additive pair relations inside an interval of length two to the power n minus two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001131 (AIM-COMBINATORICS-0256), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 3.19. Original workshop question underlying AIM-COMBINATORICS-0256. https://aimath.org/WWN/additivecomb/additivecomb.pdf.