A fresh look at the notion of normality

Abstract

Let G be a countable cancellative amenable semigroup and let (Fn) be a (left) Følner sequence in G. We introduce the notion of an (Fn)-normal element of {0,1}G. When G = (N,+) and Fn={1,2,...,n}, the (Fn)-normality coincides with the classical notion. We prove that: ∙ If (Fn) is a Følner sequence in G, such that for every α∈(0,1) we have ∑nα|Fn|<∞, then almost every x∈{0,1}G is (Fn)-normal. ∙ For any Følner sequence (Fn) in G, there exists an Cham\-per\-nowne-like (Fn)-normal set. ∙ There is a natural class of "nice" Følner sequences in (N,×). There exists a Champernowne-like set which is (Fn)-normal for every nice Følner \sq. ∙ Let A⊂N be a classical normal set. Then, for any Følner sequence (Kn) in (N,×) there exists a set E of (Kn)-density 1, such that for any finite subset {n1,n2,…,nk}⊂E, the intersection A/n1∩A/n2∩…∩A/nk has positive upper density in (N,+). As a consequence, A contains arbitrarily long geometric progressions, and, more generally, arbitrarily long "geo-arithmetic" configurations of the form {a(b+ic)j,0≤i,j≤k}. ∙ For any Følner \sq\ (Fn) in (N,+) there exist uncountably many (Fn)-normal Liouville numbers. ∙ For any nice Følner sequence (Fn) in (N,×) there exist uncountably many (Fn)-normal Liouville numbers.