Normal sets in amenable semigroups

Vitaly Bergelson, Tomasz Downarowicz and Michal Misiurewicz


Let (Fn) be a (left) Folner sequence in a countable amenable semigroup G which is embeddable in a group. We introduce the notion of (Fn)-normal set in G and (Fn)-normal sequence in {0,1}G. When G = (N,+) and Fn = {1,2,...,n}, the (Fn)-normality coincides with the classical notion. We prove several results about this notion, for example: We also investigate and juxtapose combinatorial and Diophantine properties of normal sets in semigroups (N,+) and (N,×). Below is a sample of results that we obtain: