The polynomial Bogolyubov conjecture

Adding a set to itself a few times smooths it out: if $A$ occupies a fraction $\alpha$ of the vector space $\mathbb{F}_2^n$, then a sum of a few copies of $A$ contains an entire affine subspace of large dimension. The conjecture says that the codimension of that subspace can be as small as a constant times $\log(1/\alpha)$, which would be best possible. It is stronger than the polynomial Freiman–Ruzsa conjecture, proved in 2023, and is not reached by the methods of that proof.

Problem

For $A \subseteq \mathbb{F}_2^n$ write $kA = \{a_1 + \cdots + a_k : a_i \in A\}$, and let $\alpha = |A|/2^n$ be the density of $A$.

Conjecture. There is an absolute constant $C$ such that for every $n$ and every nonempty $A \subseteq \mathbb{F}_2^n$, the set $10A$ contains a coset $x + V$ of a subspace $V$ with

$$\dim V \ge n - C \log(1/\alpha).$$

Green notes that $10A$ could perhaps be replaced by $3A$, and that $10A$ suffices for most applications. A related question: must $2A$ contain 99% of some coset of a subspace of codimension $O(\log(1/\alpha))$? The bound would be optimal, since a subspace of codimension $\log_2(1/\alpha)$ has density $\alpha$ and all its sumsets equal itself.

Background

Bogolyubov's lemma, a classical Fourier-analytic argument, shows that $4A = 2A - 2A$ contains a large subspace; the question is how large, as a function of the density. The conjecture is often called the polynomial Bogolyubov conjecture, by analogy with the polynomial Freiman–Ruzsa (Marton) conjecture, which it implies. Gowers, Green, Manners and Tao proved Marton's conjecture in 2023, but Green notes that their methods do not address this stronger statement. Analogues for subsets of $\{1, \dots, N\}$, with Bohr sets in place of subspaces, are presumably at least as hard.

What is known

• Sanders (2012): $4A$ contains a coset of a subspace of codimension $O(\log^{4 + o(1)}(1/\alpha))$.
• Kościuszko (2023), building on an unpublished idea of Konyagin: for every $\eta > 0$ there is $m$ such that $mA - mA$ contains a subspace of codimension $O(\log^{3 + \eta}(1/\alpha))$. In $\mathbb{F}_2^n$, $mA - mA = 2mA$.
• The polynomial Freiman–Ruzsa conjecture, a consequence, was proved by Gowers, Green, Manners and Tao (2023; Annals of Mathematics 2025).

What would count as progress

• A proof that some fixed sumset $kA$ contains a subspace of codimension $O(\log^{c}(1/\alpha))$ with $c < 3$, or of the full conjecture.
• A proof or disproof of the 99% version for $2A$.
• Counterexamples showing that $3A$ (or $10A$) cannot work with codimension $O(\log(1/\alpha))$. For explicit sets in small dimensions, the largest coset inside $kA$ can be found by computer.
• A formal proof of the Lean statement green_50 in GreensOpenProblems/50.lean (google-deepmind/formal-conjectures).

References

1. T. Sanders, On the Bogolyubov–Ruzsa lemma, Anal. PDE 5 (2012), 627–655; arXiv:1011.0107.
2. T. Kościuszko, Counting solutions to invariant equations in dense sets, arXiv:2306.08567 (2023), Theorem 9.
3. W. T. Gowers, B. Green, F. Manners and T. Tao, On a conjecture of Marton, Ann. of Math. 201 (2025), 515–549; arXiv:2311.05762.

Source

Imported from Ben Green's 100 open problems 50, curated by Ben Green. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.