Exact structure inside a two-fold sumset

Definitions and mathematical statement

For subsets of $\mathbb F_2^n$, sums are taken coordinatewise modulo two, and density means cardinality divided by $2^n$. Given finite nonempty sets $S_1,\ldots,S_\ell\subseteq\mathbb F_2^n$, write \[ S_1+\cdots+S_\ell =\{s_1+\cdots+s_\ell:s_i\in S_i\text{ for every }i\}. \] Repeated sums are counted only once. The sets $S_i$ are otherwise arbitrary and are not required to be subspaces or to have the same cardinality.

The problem asks whether there exist absolute constants $c>0$, $C\geq1$, and $D\geq1$ such that for every $n\geq1$ and every $A\subseteq\mathbb F_2^n$ of density at least $0.01$, one can find $B$ and $S_1,\ldots,S_\ell$ with \[ |B|\geq c2^n,\qquad B+B\subseteq A+A,\qquad B=S_1+\cdots+S_\ell, \] where $1\leq\ell\leq Dn^C$ and $1\leq|S_i|\leq Dn^C$ for all $i$. All constants must be uniform over the dimension and over $A$. The containment is exact, not a high-density or approximate containment. The source's fixed density is interpreted as a lower density threshold, since exact density $1/100$ is impossible in a vector space of cardinality $2^n$. The desired conclusion supplies a dense iterated sum of sets of polynomial size inside the additive structure already forced by $A$.

Short English statement

Does the sumset of every fixed-density binary set contain the double sumset of a dense set built from polynomially many small summands?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000907 (AIM-COMBINATORICS-0032), 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 2.55. Original workshop question underlying AIM-COMBINATORICS-0032. http://aimpl.org/addcombapp/2/.