Remarks on coarse triviality of asymptotic Assouad–Nagata dimension

We show that for a given metric space ( X , d ) of asymptotic dimension n there exists a coarsely and topologically equivalent hyperbolic metric d ′ of the form d ′ = f ∘ d such that ( X , d ′ ) is of asymptotic Assouad–Nagata dimension n. As a corollary we construct examples of spaces realising strict inequality in the logarithmic law for asdim AN of a Cartesian product. One of them may be viewed as a counterexample to a specific kind of a Morita-type theorem for asdim AN .