A fresh look at the notion of normality (2004.05058v1)

Abstract: Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{\o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of ${0,1}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = {1,2,...,n}$, the $(F_n)$-normality coincides with the classical notion. We prove that: $\bullet$ If $(F_n)$ is a F{\o}lner sequence in $G$, such that for every $\alpha\in(0,1)$ we have $\sum_n \alpha^{|F_n|}<\infty$, then almost every $x\in{0,1}^G$ is $(F_n)$-normal. $\bullet$ For any F{\o}lner sequence $(F_n)$ in $G$, there exists an Cham-per-nowne-like $(F_n)$-normal set. $\bullet$ There is a natural class of "nice" F{\o}lner sequences in $(\mathbb N,\times)$. There exists a Champernowne-like set which is $(F_n)$-normal for every nice F{\o}lner \sq. $\bullet$ Let $A\subset\mathbb N$ be a classical normal set. Then, for any F{\o}lner sequence $(K_n)$ in $(\mathbb N,\times)$ there exists a set $E$ of $(K_n)$-density $1$, such that for any finite subset ${n_1,n_2,\dots,n_k}\subset E$, the intersection $A/{n_1}\cap A/{n_2}\cap\ldots\cap A/{n_k}$ has positive upper density in $(\mathbb N,+)$. As a consequence, $A$ contains arbitrarily long geometric progressions, and, more generally, arbitrarily long "geo-arithmetic" configurations of the form ${a(b+ic)^j,0\le i,j\le k}$. $\bullet$ For any F{\o}lner \sq\ $(F_n)$ in $(\mathbb N,+)$ there exist uncountably many $(F_n)$-normal Liouville numbers. $\bullet$ For any nice F{\o}lner sequence $(F_n)$ in $(\mathbb N,\times)$ there exist uncountably many $(F_n)$-normal Liouville numbers.