On thin carpets for doubling measures

We study subsets of ${\mathbb{R}}^d$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of ${\mathbb{R}}^d$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also prove that a self-affine set that satisfies $OSCH$ (open set condition with holes) is thin for isotropic doubling measures. For doubling measures, we prove that Bara'nski carpets are thin for doubling measures.