A new construction of universal spaces for asymptotic dimension

For each $n$, we construct a separable metric space $\mathbb{U}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($\mathop{asdim}$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($\mathop{udim}$) at most $n$. Thus, $\mathbb{U}_n$ serves as a universal space for dimension $n$ in both the large-scale and infinitesimal topology. More precisely, we prove: $$\mathop{asdim} \mathbb{U}_n = \mathop{udim} \mathbb{U}_n = n$$ and such that for each separable metric space $X$, a) if $\mathop{asdim} X \leq n$, then $X$ is coarsely equivalent to a subset of $\mathbb{U}_n$; b) if $\mathop{udim} X \leq n$, then $X$ is uniformly homeomorphic to a subset of $\mathbb{U}_n$.