OP-00987 Open

The polynomial Bogolyubov conjecture over F_2^n

Let $A \subset \mathbb{F}_2^n$ have density $\alpha$, and write $10A = A + \dots + A$ (ten summands). The statement below may well hold with $3A$ in place of $10A$. It would imply the polynomial Freiman–Ruzsa theorem of Gowers, Green, Manners and Tao, whose methods do not reach it, and is probably strictly stronger. Sanders proved it with dimension $n - O(\log^{4 + o(1)}(1/\alpha))$, and Kościuszko, building on unpublished work of Konyagin, showed that for every $\eta > 0$ some $mA - mA$ contains a subspace of dimension $n - O(\log^{3 + \eta}(1/\alpha))$.

Problem. Does $10A$ contain a coset of a subspace of dimension at least $n - O(\log(1/\alpha))$?

Source: B. Green, Finite field models in additive combinatorics (2005), Question 10.4 (asked there for 2A − 2A); known as the polynomial Bogolyubov conjecture. Listed as Problem 50 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: NT
Keywords: Bogolyubov's lemma, finite field model, polynomial Freiman–Ruzsa conjecture, sumsets
Literature checked: Oct. 26, 2026 by clemens-admin

References

@incollection{green2005ffm,
  author = {Green, Ben},
  title = {Finite field models in additive combinatorics},
  booktitle = {Surveys in Combinatorics 2005},
  series = {London Mathematical Society Lecture Note Series},
  volume = {327},
  pages = {1--28},
  publisher = {Cambridge University Press},
  year = {2005},
  doi = {10.1017/CBO9780511734885.002},
  note = {Question 10.4}
}

@article{sanders2012bogolyubov,
  author = {Sanders, Tom},
  title = {On the {B}ogolyubov--{R}uzsa lemma},
  journal = {Analysis \& PDE},
  volume = {5},
  number = {3},
  pages = {627--655},
  year = {2012},
  doi = {10.2140/apde.2012.5.627}
}

@misc{kosciuszko2023,
  author = {Ko{\'s}ciuszko, Tomasz},
  title = {Counting solutions to invariant equations in dense sets},
  year = {2023},
  eprint = {2306.08567},
  archivePrefix = {arXiv},
  note = {Theorem 9}
}

@article{gowers2025marton,
  author = {Gowers, W. T. and Green, Ben and Manners, Freddie and Tao, Terence},
  title = {On a conjecture of {M}arton},
  journal = {Annals of Mathematics (2)},
  volume = {201},
  number = {2},
  pages = {515--549},
  year = {2025},
  doi = {10.4007/annals.2025.201.2.5}
}

@misc{raghavan2025freiman,
  author = {Raghavan, Rushil},
  title = {Improved bounds for the {F}reiman--{R}uzsa theorem},
  year = {2025},
  eprint = {2512.11217},
  archivePrefix = {arXiv}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 50; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00987, "The polynomial Bogolyubov conjecture over F_2^n", clopen.solutions.