Remarks on coarse triviality of asymptotic Assouad-Nagata dimension (Q2446496)

In this paper, given any metric space \((X, d)\) with \(\text{asdim}(X, d) = n\), the author constructs a coarsely and topologically equivalent hyperbolic metric \(d^{\prime}\) of the form \(d^{\prime} = c\circ d\), where \(c: [0,\infty)\to[0,\infty)\) is some concave function with \(c(r)=0\) iff \(r=0\), such that the asymptotic Assouad-Nagata dimension \(\text{asdim}_{AN} (X,d^{\prime}) = n\). The valuable point of the theorem is the direct construction of the new metric from a given metric. This allows one to carry some properties of \(\text{asdim}\) for the original metric to \(\text{asdim}_{AN}\) for the new metric. For example, a counterexample to the Morita-type theorem for \(\text{asdim}_{AN}\) is given: there exist metric spaces \(X\) and \(R\), where \(R\) is coarsely equivalent to \({\mathbb R}\), such that \(\text{asdim}_{AN} X \times R < \text{asdim}_{AN} X + \text{asdim}_{AN} R\).