Revisiting Asymptotic-Type Dimension Bounds through Combinatorial Approaches

We present an alternative probabilistic proof for the sharp Assouad--Nagata dimension bound of a doubling metric space. In addition, we explore some partial rigidity results and applications to scalar curvature. A significant technical tool in our argument is the concept of padded decomposition, which originates in computer science and has been extended to general separable metric spaces by us. Along the way, we extend the sharp upper bound on the asymptotic dimension of graphs with polynomial growth to noncollapsed locally compact metric measure spaces with polynomial volume growth. This sheds light on broader applications of probabilistic methods in metric geometry.