An ergodic correspondence principle, invariant means and applications

Abstract

A theorem due to Hindman states that if $E$ is a subset of $\mathbb{N}$ with $d^{\ast }(E)>0$, where $d^{\ast }$ denotes the upper Banach density, then for any $\epsilon >0$ there exists $N\in \mathbb{N}$ such that $d^{\ast }\left({\bigcup }_{i=1}^N(E-i)\right)>1-\epsilon$. Curiously, this result does not hold if one replaces the upper Banach density $d^{\ast }$ 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.

32 pages