Polish群中子集的Packing指数
Packing index of subsets in Polish groups
摘要：
对于波兰群$G$的一个子集$A$，我们研究了（几乎）填充指数$\ind_P(A)$（resp. $A$的基数（$\Ind_P(A)$），等于子集$S\subset G$的基数$|S|$的上确界，使得移位族$\{xA\}_{x\in S}$（几乎）不相交（在这个意义上，对于任何不同的点$x,y\in S$，$|xA\cap yA|<|A|$）。具有小（几乎）填充指数的子集$A\subset G$在几何意义上是小的。我们证明了波兰群的任何$σ$ -紧子集$A$的$\ind_P(A)\in \IN\cup\{\aleph_0,\cc\}$.如果$A\subset G$是Borel，则填充指数$\ind_P(A)$和$\Ind_P(A)$不能在半区间$[\sq(Π^1_1),\cc)$中取值，其中$\sq(Π^1_1)$是某个不可数基数，在ZFC的某些模型中小于$\cc$。在每个非离散波兰阿贝尔群$G$中，我们构造了两个闭子集$A,B\subset G$，其中有$\ind_P(A)=\ind_P(B)=\cc$和$\Ind_P(A\cup B)=1$，然后应用这个结果证明了对于任意给定的基数$κ\in[4,\cc]$，$G$包含无处稠密的Haar空子集$C\subset G$，其中有$\ind_P(C)=\Ind_P(C)=κ$.
英文摘要：
For a subset $A$ of a Polish group $G$, we study the (almost) packing index $\ind_P(A)$ (resp. $\Ind_P(A)$) of $A$, equal to the supremum of cardinalities $|S|$ of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|A|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are small in a geometric sense. We show that $\ind_P(A)\in \IN\cup\{\aleph_0,\cc\}$ for any $σ$-compact subset $A$ of a Polish group. If $A\subset G$ is Borel, then the packing indices $\ind_P(A)$ and $\Ind_P(A)$ cannot take values in the half-interval $[\sq(Π^1_1),\cc)$ where $\sq(Π^1_1)$ is a certain uncountable cardinal that is smaller than $\cc$ in some models of ZFC. In each non-discrete Polish Abelian group $G$ we construct two closed subsets $A,B\subset G$ with $\ind_P(A)=\ind_P(B)=\cc$ and $\Ind_P(A\cup B)=1$ and then apply this result to show that $G$ contains a nowhere dense Haar null subset $C\subset G$ with $\ind_P(C)=\Ind_P(C)=κ$ for any given cardinal number $κ\in[4,\cc]$.
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