Some basic properties of Assouad dimenison

Let $E$ be a subset of a metric sapce $(X, d)$ and $r>0.$ The set $E$ is called $r$-discrete if $d(x,y)\ge r$ whenever $x,y\in E, x\not=y.$ Let $N_r(B(x,R))$ denotes the cardinality of a maximal $r$-discrete subset of $B(x,R)$. A metric space $X$ is $s$-homogeneous, where $s\ge 0$, if there exists a constant $C\ge 1$ such that

$$N_r(B(x,R))\le C(\frac{R}{r})^s$$ 

for any $0<r<R<diam X$ and $x\in X.$ The Assoaud dimension of a metric space $X$ is defined by

$$\dim_AX=\inf\{s\ge 0: \text{$X$ is $s$-homogeneous}\}.$$

 

The following are some basic properties of Assouad dimension.

 

1. If $\rho$ is another metric on $X$ with $c_1d\le \rho \le c_2 d$ for some $0<c_1\le c_2,$ then $\dim_A (X,\rho)=\dim_A(X,d);$

2. If $E\subset X$, then $\dim_A E\le \dim_A X;$ 

3. $\dim_A \overline{E}=\dim_A E$, where $\overline{E}$ denotes the closure of $E;$

4. (finite stability) $\dim_A (\cup_{i=1}^n X_i)=max_i \dim_A X_i;$

5. If $X\subset R^n$ and $X$ has interior points, then $\dim_A X=m;$

6. If $X$ is bounded, then $\dim_H X\le \dim_A X$, and we always have $\dim_P X\le \dim_A X.$

7. If $x$ carries a Ahlfors $\alpha$-regular measure, i.e., there exist a Borel mesure supported on $X$ and  a constant $C<\infty$such that

$$C^{-1} r^{\alpha}\le \mu(B(x,r)\cap X)\le C r^{\alpha}$$

for any $x\in X$ and $0<r<diam X.$ Then $\dim_A X=\alpha.$

Proof.  For the proofs of Claims 1.-6., see Luukkainen's paper ''Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, Theorem A.5; For the proof of Claim 7., see Theorem 3.8 in the paper "Equi-homogeneous, Assouad dimension and non-autonomous dynamics".