Groups, Geometry, and Dynamics

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Volume 11, Issue 2, 2017, pp. 585–612
DOI: 10.4171/GGD/409

Published online: 2017-06-26

Small doubling in ordered groups: generators and structure

Gregory A. Freiman[1], Marcel Herzog[2], Patrizia Longobardi[3], Mercede Maj[4], Alain Plagne[5] and Yonutz V. Stanchescu[6]

(1) Tel Aviv University, Israel
(2) Tel Aviv University, Israel
(3) Università di Salerno, Fisciano (Salerno), Italy
(4) Università di Salerno, Fisciano (Salerno), Italy
(5) Ecole polytechnique, Palaiseau, France
(6) The Open University of Israel, Raanana, and Afeka Academic College, Tel Aviv, Israel

We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman’s $3k-3$ and $3k-2$ theorems in the integers and several further generalizations.

Keywords: Inverse problems, small doubling, nilpotent groups, ordered groups

Freiman Gregory, Herzog Marcel, Longobardi Patrizia, Maj Mercede, Plagne Alain, Stanchescu Yonutz: Small doubling in ordered groups: generators and structure. Groups Geom. Dyn. 11 (2017), 585-612. doi: 10.4171/GGD/409