关于无穷子集的稠密性I
On density of infinite subsets I
摘要：
设$Y$是紧度量空间，$G$是作用于$Y$上的变换群.对于任意的无限子集$A\subset Y$，我们利用Hausdorff半度量$d^H$研究了$g\in G$的$gA$的密度和集合$\displaystyle{\bigcup_{g\in G_n}gA}$的数量密度.证明了对任意整数$n\ge 2$，$ε>0$，任意无限子集$A\subset \mathbb T^n$，存在一个$g\in SL(n,\mathbb Z)$使得$gA$是$ε$ -稠密的.我们还表明，对于任何无限子集$A\subset [0,1]$，通用旋转和通用3-IET，$$\liminf_nn\cdot d^H\left(\bigcup_{k=0}^{n-1}T^kA,[0,1]\right)=0.$$
英文摘要：
Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\subset Y$, we study the density of $gA$ for $g\in G$ and quantitative density of the set $\displaystyle{\bigcup_{g\in G_n}gA}$ by the Hausdorff semimetric $d^H$. It is proven that for any integer $n\ge 2$, $ε>0$, any infinite subset $A\subset \mathbb T^n$, there is a $g\in SL(n,\mathbb Z)$ such that $gA$ is $ε$-dense. We also show that, for any infinite subset $A\subset [0,1]$, for generic rotation and generic 3-IET, $$\liminf_nn\cdot d^H\left(\bigcup_{k=0}^{n-1}T^kA,[0,1]\right)=0.$$
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