On asymptotic Assouad–Nagata dimension

For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad–Nagata dimension AN-asdimX of X coincides with the covering dimension dim(νLX) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X×R)=AN-asdimX+1. We note that the similar equality for Gromov's asymptotic dimension asdim generally fails to hold [A. Dranishnikov, Cohomological approach to asymptotic dimension, Preprint, 2006].Additionally we construct an injective map from the asymptotic cone without the basepoint to the sublinear Higson corona.