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Register a new userA fresh look at the notion of normality
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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 $alphain(0,1)$ we have $sum_n alpha^{|F_n|}<infty$, then almost every $xin{0,1}^G$ 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 $Asubsetmathbb N$ 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 $A/{n_1}cap A/{n_2}capldotscap A/{n_k}$ 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 ${a(b+ic)^j,0le i,jle k}$. $bullet$ 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.
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