  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.
