Papers › Marton's Conjecture in abelian groups with bounded torsion
Marton's Conjecture in abelian groups with bounded torsion
W. T. Gowers, Ben Green, Freddie Manners, Terence Tao
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We prove a Freiman--Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let G be an abelian group of torsion m (meaning mg=0 for all g ∈G) and suppose that A is a non-empty subset of G with |A+A| ≤K|A|. Then A can be covered by at most (2K)^(O(m³)) translates of a subgroup of H ≤G of cardinality at most |A|. The argument is a variant of that used in the case G = 𝐅₂ⁿ in a recent paper of the authors.
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