High density piecewise syndeticity of product sets in amenable groups

M. Beiglb\"ock, V. Bergelson, and A. Fish proved that if $G$ is a countable amenable group and $A$ and $B$ are subsets of $G$ with positive Banach density, then the product set $AB$ is piecewise syndetic. This means that there is a finite subset $E$ of $G$ such that $EAB$ is thick, that is, $EAB$ contains translates of any finite subset of $G$. When $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a F\o lner sequence) of the set of witnesses to the thickness of $% EAB$. When $G=\mathbb{Z}^d$, this result was first proven by the current set of authors using completely different techniques.