On thin carpets for doubling measures

Abstract

We study subsets of R d which are thin for doubling measures or isotropic doubling measures. We show that any subset of 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 Barański carpets are thin for doubling measures.