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Marton’s conjecture in abelian groups with bounded torsion
[La conjecture de Marton pour des groupes abéliens à torsion bornée]
W. Timothy Gowers1 ; Ben J. Green2 ; Freddie Manners3 ; Terence Tao4
1Collège de France, 11, place Marcelin-Berthelot, 75231 Paris 05, France, and Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
2Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6QW, UK
3Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, CA 92093-0112, USA
4Math Sciences Building, 520 Portola Plaza, Box 951555, Los Angeles, CA 90095, USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 1, pp. 1-33

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 $g \in G$) and suppose that $A$ is a non-empty subset of $G$ with $|A+A| \le K|A|$. Then $A$ can be covered by at most $(2K)^{O(m^3)}$ translates of a subgroup $H \le G$ of cardinality at most $|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$ in a recent paper of the authors.

Nous démontrons un théorème de type Freiman–Ruzsa avec des bornes polynomiales pour des groupes abéliens arbitraires à torsion bornée, prouvant ainsi (en toute généralité) une conjecture de Marton. Plus précisément, soit $G$ un groupe abélien de torsion $m$ (c’est-à-dire $mg=0$ pour tout $g$ dans $G$) et soit $A$ un sous-ensemble non vide de $G$ tel que $|A+A| \le K|A|$. Alors $A$ peut être couvert par au plus $(2K)^{O(m^3)}$ translatés d’un sous-groupe $H \le G$ de taille au plus $|A|$. L’argument est une variante de celui utilisé dans le cas $G = \mathbf{F}_2^n$ dans un article récent des auteurs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1839
Classification : 10X99, 14A12, 11L05
Keywords: Freiman-Ruzsa, Marton’s conjecture, sumsets, entropy
Mots-clés : semblable banalité, autosimilarité logarithmique, loi de Gauß

W. Timothy Gowers  1   ; Ben J. Green  2   ; Freddie Manners  3   ; Terence Tao  4

1 Collège de France, 11, place Marcelin-Berthelot, 75231 Paris 05, France, and Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
2 Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6QW, UK
3 Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, CA 92093-0112, USA
4 Math Sciences Building, 520 Portola Plaza, Box 951555, Los Angeles, CA 90095, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Marton{\textquoteright}s conjecture in abelian groups with bounded torsion},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1--33},
     year = {2026},
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W. Timothy Gowers; Ben J. Green; Freddie Manners; Terence Tao. Marton’s conjecture in abelian groups with bounded torsion. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 1, pp. 1-33. doi: 10.5802/afst.1839

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