Juxtaposing $d^*$ and $\bar{d}$

We take a close look at two versions of Furstenberg's correspondence principle which establish a link between the combinatorial properties of large sets in an amenable group $G$ with the properties of measure-preserving $G$-actions. Our analysis reveals that an "ergodic" version of Furstenberg's correspondence principle which uses the upper Banach density ($d^*$) is better suited for applications than the version which involves the upper density, $\bar{d}$. For example, we show that an ergodic version of Furstenberg's correspondence principle allows one to obtain an easy proof of a general version of a result of Hindman, which states that if $E$ is a subset of $\mathbb{N}$ with $d^*(E)>0$, then for any $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that $d^*\left(\bigcup_{i=1}^N(E-i)\right) > 1-\varepsilon$ (this result does not hold if one replaces the upper Banach density $d^*$ with the upper density $\bar{d}$). We also establish a new characterization of amenable minimally almost periodic groups.