Ahlfors regular spaces have regular subspaces of any dimension

We characterize Q-dimensional Ahlfors regular spaces among trees’ boundaries and show how to construct, for each 0 < α < Q, an α-regular subspace. As an application, we give an alternative simple proof of the existence of α-regular subspaces of a Q-dimensional complete Ahlfors regular metric space (X, ρ), which was proved in [8].