Approximate subloops and Freiman's theorem in finitely generated commutative moufang loops

Fix a parameter $K\geqslant 1$. A $K$-approximate subgroup is a finite set $A$ in a group $G$ which contains the identity, is symmetric and such that $A.A$ can covered by $K$ left translates of $A$. This article deals with the generalisation of the concept of approximate groups in the case of loops which we call approximate loops and the description of $K$-approximate subloops when the ambient loop is a finitely generated commutative moufang loop. Specifically we have a Freiman type theorem where such approximate subloops are controlled by arithmetic progressions defined in the commutative moufang loops.

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