2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230348Jin proved that whenever $A$ and $B$ are sets of positive upper density in $\Z$, $A+B$ is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains $\Z^d$. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or -- depending on the notation -- "productsets") are piecewise Bohr, a result which for $G=\Z$ was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group $G$, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.Dynamical Systemstext