  A theorem due to Hindman states that if $E$ is a subset of $\mathbb{N}$ with
$d^*(E)>0$, where $d^*$ denotes the upper Banach density, then for any
$\varepsilon>0$ there exists $N \in \mathbb{N}$ such that
$d^*\left(\bigcup_{i=1}^N(E-i)\right) > 1-\varepsilon$. Curiously, this result
does not hold if one replaces the upper Banach density $d^*$ with the upper
density $\bar{d}$. Originally proved combinatorially, Hindman's theorem allows
for a quick and easy proof using an ergodic version of Furstenberg's
correspondence principle. In this paper, we establish a variant of the ergodic
Furstenberg's correspondence principle for general amenable (semi)-groups and
obtain some new applications, which include a refinement and a generalization
of Hindman's theorem and a characterization of countable amenable minimally
almost periodic groups.
