An ergodic correspondence principle, invariant means and applications

2020/03/06 by Bergelson, Vitaly, Moragues, Andreu Ferré
#05D10 (Secondary) #37A15 (Primary) 28D15 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.03029

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^*(\bigcup_i=1^N(E-i)) > 1-ε. Curiously, this result does not hold if one replaces the upper Banach density d^* with the upper density 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.

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