Higher additive energy and quantitative stability of the Holder defect

Definitions and mathematical statement

Let $A$ be a finite nonempty subset of an abelian group $G$. For $j\geq2$, define its higher additive energies and their normalizations by \[ E_{2j}(A)=|\{(a_1,\ldots,a_{2j})\in A^{2j}: a_1+\cdots+a_j=a_{j+1}+\cdots+a_{2j}\}|, \qquad \rho_{2j}(A)=E_{2j}(A)/|A|^j. \] These are counts of ordered tuples, allowing repetitions. For an integer $m\geq1$, $mA$ denotes the set of sums of $m$ elements of $A$, again allowing repetitions.

Fix an integer $n\geq2$ and a real $\varepsilon\in(0,1)$. Does there exist a pair of positive constants $c,c'$, depending only on $n,\varepsilon$, such that the following assertion holds for every $G,A$ and every $K>1$? If \[ \rho_{2n+2}(A)^{(n-1)/n}<K\rho_{2n}(A), \] then some nonempty $H\subseteq(n-1)A$ satisfies \[ |H|\geq\frac{\rho_{2n}(A)^{1/(n-1)}}{K^c}, \qquad E_4(H)\geq\frac{|H|^3}{|A|^{\varepsilon}K^{c'}}. \] Thus near equality in the comparison between successive normalized energies should produce a relatively large set with substantial ordinary additive energy. The constants must not depend on $A$, its ambient group, or $K$. The factor $|A|^{\varepsilon}$ in the energy denominator is retained: the source discussion explicitly corrects a formulation that had omitted this loss. No claim that the extracted set is a subgroup is part of the question.

Short English statement

Does slow growth between successive higher additive energies force a large, energetically structured subset of an iterated sumset?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000908 (AIM-COMBINATORICS-0033), 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 2.6. Original workshop question underlying AIM-COMBINATORICS-0033. http://aimpl.org/addcombapp/2/.