We introduce the notion of weak expander spaces.

\[DefinitionWeakExpander\] A generalized box space $X = \bigsqcup_{m = 1}^\infty X_m$ is called a sequence of weak expander spaces if there exist a controlled set $T$ and a positive number $c$ satisfying the following condition: for every controlled set $\widetilde{T}$, $$\begin{aligned}
\liminf_{m \rightarrow \infty} \left(
\inf \left\{ \frac{ \sharp(T[Y]) }{ \sharp(Y) } \ \colon\ 
\emptyset \neq Y \subseteq X_m, Y \textrm{\ is\ a } 
\widetilde{T} \textrm{-bounded\ set} \right\} \right) > 1 + c.\end{aligned}$$

Consider the case that $T$ contains the diagonal set. Then the set $T[Y]$ is a kind of $Y$’s neighborhood and we regard the term $\sharp(T[Y]) / \sharp (Y)$ as the expansion of $Y$. The condition in Definition \[DefinitionWeakExpander\] means that the expansion of $Y$ is uniformly greater than $1 + c$ if $m$ is large enough and if $Y$ is a $\widetilde{T}$-bounded set. We first observe that the cardinality of $X_m$ diverges, like an expander sequence.

\[LemmaDiverge\] If a generalized box space $X = \bigsqcup_{m = 1}^\infty X_m$ is a sequence of weak expander spaces, then $\lim_{m \rightarrow \infty} \sharp(X_m) = \infty$.

Take a controlled set $T$ and a positive number $c$ in Definition \[DefinitionWeakExpander\]. We may assume that 