On Sierpiński carpets and doubling measures

Abstract

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.

A function whose graph has positive doubling measure

“…The only thing left to check is that we actually have a doubling measure as claimed. For this purpose we rephrase Lemma 2 from [2]. After this it is easy to check that the construction at hand satisfies the assumptions of the lemma.…”

“…For instance one could study the connection between the doubling constant and the modulus of continuity in this problem. To our knowledge not much is known on such questions beyond the obviously thin graphs of functions with finite lip as defined in (2), and the example provided in this note. For instance the following natural question is still open:…”