$\Gamma_K \subseteq X^2$ by $T_K = \{ (g x, x) \ |\ g \in K, x \in X\}$. We say that a subset $T \subseteq X \times X$ is controlled if there exists a finite subset $K \subseteq G$ such that $T \subseteq T_K$. This coarse structure on $X$ is uniformly locally finite.

Generalized box space
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The term ‘box space’ was introduced in [@RoeLectureNote Definition 11.24]. It is a sequence of finite quotient groups of a residually finite group $G$. Let $H_1 \supseteq H_2 \supseteq \cdots$ be finite index normal subgroups of $G$ whose intersection $\bigcap_{m=1}^\infty H_m$ is $\{1_G\}$. A box space is the disjoint union $\square G = G / H_1 \bigsqcup G/H_2 \bigsqcup \cdots$ of finite quotient groups. The natural left translation action of $G$ gives a uniformly locally finite coarse structure on $\square G$.

Box spaces give interesting examples related to property A as follows:

-   Guentner observed that $\square G$ has property A if and only if $G$ is amenable ([@RoeLectureNote Proposition 11.39]).

-   It has been already pointed out by Willett that the uniform Roe algebra $C^*_\mathrm{u}( \square G )$ is not even exact if $G$ is not amenable (see the last sentence of 