论文标题
重新看一下正常的概念
A fresh look at the notion of normality
论文作者
论文摘要
令$ g $为可数的取消式正序的半群，让$（f_n）$为$ g $中的følner序列。我们介绍了$（f_n）$ - $ \ {0,1 \}^g $的普通元素的概念。当$ g $ = $（\ Mathbb n，+）$和$ f_n = \ {1,2，...，n \} $，$（f_n）$ - 正态性与经典概念相吻合。我们证明： $ \ bullet $如果$（f_n）$是$ g $中的følner序列，以至于我们有$ \sum_nα^{| f_n |} <\ infty $的每一个$α\ in（0,1）$，那么几乎每个$ x \ in \ in \ in \ in \ in \ {0,1 \}^g $（f_ is $（f _ in $）$（f_n）$（f_n）$ - 正常。 $ \ bullet $用于$ g $中的任何følnersequence $（f_n）$，存在一个cham \ -per \ -nowne-nowne like $（f_n）$ - 普通套件。 $ \ bullet $ $（\ mathbb n，\ times）$中有天然的“ nice”følner序列。存在一个类似Champernowne的套件，即$（f_n）$ - 对于每个NICEFølner\ sq来说，正常。 $ \ bullet $让$ a \ subset \ mathbb n $是经典的普通集。然后，对于任何Følner序列$（k_n）$（\ Mathbb n，\ times）$，存在$ e $（k_n）$ - 密度 - 密度$ 1 $，因此对于任何有限的子集$ \ {n_1，n_1，n_2，n_2，n_k \} a/{n_2} \ cap \ ldots \ cap a/{n_k} $在$（\ mathbb n，+）$中具有正密度。结果，$ a $包含任意长的几何进程，更一般而言，更一般而言，是$ \ {a（b+ic）^j，0 \ le i，j \ le k \} $的$ \ {a（b+ic）^j，0的“地理算术”配置。 任何Følner\ sq \ $（f_n）$（\ Mathbb n，+）$的$ \ bullet $存在许多$（f_n）$ - 普通liouville数字。 $ \ bullet $用于$（\ mathbb n，\ times）$中的任何nicefølnersequence $（f_n）$，存在许多$（f_n）$ - 普通的liouville数字。
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.
