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6 changes: 3 additions & 3 deletions constants/18a.md
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## Description of constant

$C_{18}$ is the least constant such that, whenever $A$ is a subset of $\mathbb F\_{2}^n$ with $\lvert A+A\rvert \leq K\lvert A\rvert$, then $A$ can be covered by $K^{C_{18}+o(1)}$ cosets of a subspace of cardinality at most $\lvert A\rvert$, where the limit $o(1)$ is with respect to the limit $K \to \infty$.
$C_{18}$ is the least constant such that, whenever $A$ is a subset of $\mathbb F_2^n$ with $\lvert A+A\rvert \leq K\lvert A\rvert$, then $A$ can be covered by $K^{C_{18}+o(1)}$ cosets of a subspace of cardinality at most $\lvert A\rvert$, where the limit $o(1)$ is with respect to the limit $K \to \infty$.

## Known upper bounds

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## Additional comments and links

- Conjectured to be finite by Katalin Marton, as recorded in [R1999]. It is the special case of the Polynomial Freiman-Ruzsa (PFR) conjecture when the ambient group is a vector space over the field $\mathbb F\_{2}$. (The precise formulation of the PFR conjecture in the case of unbounded torsion is still not fully settled.)
- Conjectured to be finite by Katalin Marton, as recorded in [R1999]. It is the special case of the Polynomial Freiman-Ruzsa (PFR) conjecture when the ambient group is a vector space over the field $\mathbb F_2$. (The precise formulation of the PFR conjecture in the case of unbounded torsion is still not fully settled.)
- The lower bound of 1 is not expected to be sharp.
- Surveys on this problem can be found at [G2005], [G-unpub], and [Lovett2015].

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- [G-unpub] Green, B. J. *Notes on the polynomial Freiman–Ruzsa conjecture.* Unpublished note available at https://people.maths.ox.ac.uk/greenbj/papers/PFR.pdf
- [GGMT2025] Gowers, W. T.; Green, B.; Manners, F.; Tao, T. *On a conjecture of Marton.* Annals of Mathematics, Second Series, Volume 201 (2025), Issue 2, 515–549. [arXiv:2311.05762](https://arxiv.org/abs/2311.05762)
- [Lovett2015] Lovett, S. *An Exposition of Sanders’ Quasi-Polynomial Freiman–Ruzsa Theorem.* Theory of Computing Library Graduate Surveys 6 (2015), 1–14.
- [L2024] Liao, J.-J. *Improved Exponent for Marton's Conjecture in $\mathbb F\_{2}^n$.* [arXiv:2404.09639](https://arxiv.org/abs/2404.09639) (2024).
- [L2024] Liao, J.-J. *Improved Exponent for Marton's Conjecture in $\mathbb F_2^n$.* [arXiv:2404.09639](https://arxiv.org/abs/2404.09639) (2024).
- [R1999] Ruzsa, I. Z. *An analog of Freiman’s theorem in groups.* Astérisque 258 (1999), 323–326.