A fresh look at the notion of normality
Abstract
Let G be a countable cancellative amenable semigroup and let (Fn) be a (left) Flner 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 Flner sequence in G, such that for every α∈(0,1) we have Σn α|Fn|<∞, then almost every x∈\0,1\G is (Fn)-normal. For any Flner sequence (Fn) in G, there exists an Cham\-per\-nowne-like (Fn)-normal set. There is a natural class of "nice" Flner sequences in ( N,×). There exists a Champernowne-like set which is (Fn)-normal for every nice Flner . Let A⊂ N be a classical normal set. Then, for any Flner 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 Flner \ (Fn) in ( N,+) there exist uncountably many (Fn)-normal Liouville numbers. For any nice Flner sequence (Fn) in ( N,×) there exist uncountably many (Fn)-normal Liouville numbers.