Relaxed Subspaces-in-Sumsets Conjecture (99% containment)

Establish that for any fixed α > 0 and A ⊆ F2^n of density at least α, the sumset A + A contains 99% of the points of an affine subspace of codimension O(log(1/α)).

Background

The original question asks whether A + A contains an entire subspace of codimension O(√n); this is known to be tight if true, and current best bounds give subspaces of dimension roughly (α/ln 2)·n or codimension about n/log2(1/ε) near the threshold.

A relaxed variant asks only for near-full containment (e.g., 99%) of a large subspace, conjecturally allowing codimension O(log(1/α)), which would represent a substantial strengthening aligned with Polynomial Bogolyubov-type phenomena.

References

As noted in the remarks on the Polynomial Freiman--Ruzsa/Bogolyubov Conjectures, it is also interesting to consider the relaxed problem where we only require that A + A contains 99% of the points in a large subspace. Here it might be conjectured that the subspace can have codimension O(log(1/α)).

— Open Problems in Analysis of Boolean Functions  (1204.6447 - O'Donnell, 2012) in Main matter, problem “Subspaces in Sumsets,” last bullet