Examples of metric spaces with asymptotic property $C$

Abstract: We construct a class of metric spaces whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\omega+k$ for any $k\in\mathbb{N}$, where $\omega$ is the smallest infinite ordinal number and a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $2\omega$. Moreover, we study the relationship between asymptotic dimension growth, transfinite asymptotic dimension and finite decomposition complexity.