Let G be a group of bounded exponent, meaning that every element has order bounded by an absolute constant, and let A⊂ G satisfy |A+A|≤ K|A|. A subgroup cover is a subgroup M whose translates cover A; the size of the cover is the number of translates, and |M| denotes its cardinality. Marton's conjecture. A is exp(O(log K))-covered by a subgroup M of size at most exp(O(log K))|A|. This is the bounded-exponent formulation of the polynomial Freiman–Ruzsa conjecture and would improve the quasipolynomial covering and size bounds available in the theorem stated immediately beforehand.

Marton's conjecture

Marton’s conjecture, attributed to Katalin Marton and published by Imre Ruzsa in 1999, asks for polynomial bounds on subgroup covers of small-doubling sets. The stated source evidence concerns the abelian bounded-exponent formulation.

Known results

Sanders, 2012: obtained bounds slightly above polynomial.
Gowers, Green, Manners, and Tao, 2023: proved the conjecture in the binary-vector-space case, with roughly $K^{12}$ cosets and subgroup size at most $|A|$ .
Liao, later: reduced the binary-case exponent from $12$ to $9$ .
Tao and collaborators, December 2023: formally verified the binary-case proof in Lean.

Full abelian bounded-torsion claim

Version $2$ of a manuscript by Gowers, Green, Manners, and Tao states that for abelian groups of exponent dividing $m$ , one can cover $A$ by at most $(2K)^{O(m^3)}$ translates of a subgroup of size at most $|A|$ . This would prove the conjecture for each fixed exponent, but the supplied evidence does not independently verify the manuscript’s proof. A 2026 algorithmic paper treats this structural theorem as its underlying result.

Current status (as of September 2026): The binary-vector-space case is formally verified, while the full abelian bounded-exponent conjecture is claimed by a manuscript but remains unverified; a genuinely nonabelian interpretation remains open.

Marton's conjecture

Primary source

Progress summary

Solutions 0