A fresh look at the notion of normality

Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) Følner 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 Følner sequence in $G$, such that for every $α\in(0,1)$ we have $\sum_n α^{|F_n|}<\infty$, then almost every $x\in\{0,1\}^G$ is $(F_n)$-normal. $\bullet$ For any Følner sequence $(F_n)$ in $G$, there exists an Cham\-per\-nowne-like $(F_n)$-normal set. $\bullet$ There is a natural class of "nice" Følner sequences in $(\mathbb N,\times)$. There exists a Champernowne-like set which is $(F_n)$-normal for every nice Følner \sq. $\bullet$ Let $A\subset\mathbb N$ be a classical normal set. Then, for any Følner 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 Følner \sq\ $(F_n)$ in $(\mathbb N,+)$ there exist uncountably many $(F_n)$-normal Liouville numbers. $\bullet$ For any nice Følner sequence $(F_n)$ in $(\mathbb N,\times)$ there exist uncountably many $(F_n)$-normal Liouville numbers.

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