An ergodic correspondence principle, invariant means and applications

Abstract

A theorem due to Hindman states that if E is a subset of ℕ with d*(E) > 0, where d* denotes the upper Banach density, then for any ε > 0 there exists N ∈ ℕ such that $$d^{\ast}(\cup_{i=1}^{N}(E-i))>1-\varepsilon$$ . 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.