Polynomial Bogolyubov conjecture: large subgroup inside 2A − 2A

Determine whether, for every abelian group G of torsion m and every finite non-empty subset A ⊆ G with doubling |A + A| ≤ K|A|, the difference set 2A − 2A contains a subgroup H with cardinality at least K^{-O_m(1)}|A| (the polynomial Bogolyubov conjecture).

Background

The paper proves a polynomial Freiman–Ruzsa-type theorem (Marton’s conjecture) for abelian groups with bounded torsion m, giving polynomial bounds on the number of cosets needed to cover A by a subgroup H of size at most |A|. Prior results (e.g., Sanders and Konyagin) provided exponential-type bounds or did not guarantee H ⊆ 2A − 2A.

A stronger and long-standing objective is the polynomial Bogolyubov conjecture, which asserts that 2A − 2A already contains a subgroup H of size polynomially comparable to |A| in K (with exponent depending on m). The authors’ methods yield a weaker conclusion: they can place H inside ℓA − ℓA for some ℓ polylogarithmic in K, rather than achieving ℓ = 2 as demanded by the conjecture, and they note they could not reach this even for m = 2.