Symmetric Kneser's Theorem with Trios and $3$-Transform

Abstract: We give a new equivalent restatement and a new proof in terms of trios to the classical Kneser's theorem. In the finite case, our restatement takes the following, particularly symmetric shape: if $A$, $B$, and $C$ are subsets of a finite abelian group $G$ such that $A+B+C\ne G$, then, denoting by $H$ the period of the sumset $A+B+C$, we have $$ |A|+|B|+|C| \le |G|+|H|. $$ The proof is based on an extension of the familiar Dyson transform onto set systems containing three (or more) sets.