Property A and asymptotic dimension

TL;DR

This paper characterizes the asymptotic dimension of metric spaces using conditions similar to Property A, providing equivalent criteria involving finite subsets and controlled overlaps.

Contribution

It introduces new characterizations of asymptotic dimension in terms of Property A-like conditions, linking geometric and combinatorial properties.

Findings

Equivalent conditions for asymptotic dimension involving finite subsets and overlaps.

Connections established between Property A and asymptotic dimension.

Provides a framework for analyzing metric spaces via these properties.

Abstract

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 $R,\epsilon > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_x\Delta A_y|}{| A_x\cap A_y|} < \epsilon$ 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 $\frac{| A_x\Delta A_y|}{| A_x\cap A_y|} < \frac{1}{n+1}$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$.