On Sierpiński carpets and doubling measures

According to the size of sets in the sense of doubling measures, subsets of the Euclidean space R super(n) can be divided into six classes: very fat VF, fairly fat FF, minimally fat MF, very thin VT, fairly thin FT and minimally thin MT. Let S be a Sierpinski carpet and let C be anyone of the above classes of sets in the plane. We obtain a sufficient and necessary condition for S [setmembership] C in terms of the defining data of S.