Let E = E({n k }, {c k }) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if (n k c k ) p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and only if there are constants 0 < p < q < 1 such that (n k c k ) q < ∞ and (n k c k ) p = ∞. The classes of minimally thin uniform Cantor sets and of fairly thin uniform Cantor sets are also characterized.

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According to the size of sets in the sense of doubling measures, subsets of the Euclidean space R n can be divided into six classes: very fat V F , fairly fat F F , minimally fat M F , very thin V T , fairly thin F T and minimally thin M T . Let S be a Sierpiński carpet and let C be anyone of the above classes of sets in the plane. We obtain a sufficient and necessary condition for S ∈ C in terms of the defining data of S.

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