性质A和渐近维数
Property A and asymptotic dimension
摘要：
本文的目的是用类似于Yu的性质A的方法刻画度量空间$X$的渐近维数$asdim(X)$：若$(X,d)$是度量空间，且$n\ge 0$，则下列条件等价：[a.] $asdim(X,d)\leq n$，[b.]对于每个$R,ε> 0$，存在$S > 0$和有限非空子集$A_x\subset B(x,S)\times N$、$x\in X$，使得如果$d(x,y) < R$和$A_x$到$X$上的投影对于所有$x\in X$至多包含$n+1$元素，则$\frac{| A_xΔA_y|}{| A_x\cap A_y|} < ε$，[c.]对于每个$R > 0$，存在$S > 0$和有限的非空子集$A_x\subset B(x,S)\times N$、$x\in X$，使得如果$d(x,y) < R$和$A_x$到$X$上的投影对于所有$x\in X$至多包含$n+1$元素，则$\frac{| A_xΔA_y|}{| A_x\cap A_y|} < \frac{1}{n+1}$。
英文摘要：
The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.] $asdim(X,d)\leq n$, [b.] For each $R,ε> 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_xΔA_y|}{| A_x\cap A_y|} < ε$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$, [c.] For each $R > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_xΔA_y|}{| A_x\cap A_y|} < \frac{1}{n+1}$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
