Assouad dimensions of some special sets

Example 1. Let $E=\{a_n\}_{n\ge 1}\subset R$ satisfying the conditon : there exist $C>0$ and $0<\alpha<1$ such that

$$C^{-1}\alpha^n\le a_n\le C\alpha^n.$$

Then $\dim_AE=0.$

Proof. See Lemma 9.10 in Dimensions, Embeddings, and Attractors.

Example 2.   Let $\alpha>0, E_{\alpha}=\{0\}\cup\{\frac{1}{n^{\alpha}}\}\subset R$. Then $\dim_BE_{\alpha}=1/(1+\alpha), \dim_A E_{\alpha}=1.$

Proof. See Proposition 2.5 in Equi-homogeneity, Assouad Dimension and Non-autonomous Dynamics.