A fresh look at the notion of normality

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 $F_n=\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$, 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 ${\sum }_n{\alpha }^{\mathrm{\mid }{F}_n\mathrm{\mid }}<\mathrm{\infty }$, 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 $(\mathbb N,\times)$\mathrm{.}ThereexistsaChampernowne-likesetwhichis$(F_n)$-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 $A/{n_1}\cap A/{n_2}\cap\ldots\cap A/{n_k}$haspositiveupperdensityin$(\mathbb N,+)$. 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.