Finite products sets and minimally almost periodic groups

We characterize those locally compact, second countable, amenable groups in which a density version of Hindman's theorem holds and those countable, amenable groups in which a two-sided density version of Hindman's theorem holds. In both cases the possible failure can be attributed to an abundance of finite-dimensional unitary representations, which allows us to construct sets with large density that do not contain any shift of a set of measurable recurrence, let alone a shift of a finite products set. The possible success is connected to the ergodic-theoretic phenomenon of weak mixing via a two-sided version of the Furstenberg correspondence principle.We also construct subsets with large density that are not piecewise syndetic in arbitrary non-compact amenable groups. For countably infinite amenable groups, the symbolic systems associated to such sets admit invariant probability measures that are not concentrated on their minimal subsystems.