Remarks on coarse triviality of asymptotic Assouad–Nagata dimension

We show that for a given metric space (X,d)(X,d) of asymptotic dimension n   there exists a coarsely and topologically equivalent hyperbolic metric d′d′ of the form d′=f∘dd′=f∘d such that (X,d′)(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 asdimANasdimAN of a Cartesian product. One of them may be viewed as a counterexample to a specific kind of a Morita-type theorem for asdimANasdimAN.