A variational energy core for stronger BSG

Definitions and mathematical statement

Let $G$ be an abelian group and let $A\subset G$ be finite and nonempty. For any finite $S\subseteq G$, write \[ E(S)=|\{(a,b,c,d)\in S^4:a+b=c+d\}|, \qquad e(S)=E(S)/|S|^3 \] when $S\ne\varnothing$. All tuples are ordered and repeated entries are allowed. The sumset $S+S$ consists of distinct sums, so its cardinality measures doubling, whereas $E(S)$ measures their repeated representations.

Given a real parameter $K\geq1$ with $E(A)\geq |A|^3/K$, must there exist a nonempty subset $A'\subseteq A$ satisfying all three inequalities \[ |A'|\geq |A|/K^{100},\qquad |A'+A'|\leq K^{100}|A'|,\qquad e(A')\geq e(A)? \] The workshop further asks whether the same size and doubling conclusions can hold together with the stronger energy gain \[ e(A')\geq e(A)\left(\frac{|A|}{|A'|}\right)^{1/100}. \] These are universal assertions over $G,A,K$; the subset may depend on all of them. The first question asks that passing to a structured subset never decrease normalized additive energy. The second demands a quantified increase when the retained subset is small. The number $100$ and the exponent $1/100$ are retained from the stated formulation, rather than silently replaced by unspecified constants. The ambient group need not be finite or torsion-free, since all counts involve only the given finite sets.

Short English statement

Can the Balog--Szemerédi--Gowers subset be chosen without losing normalized additive energy, and even with a specified energy gain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000899 (AIM-COMBINATORICS-0024), 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 2.15. Original workshop question underlying AIM-COMBINATORICS-0024. http://aimpl.org/addcombapp/2/.