Fat and thin sets for doubling measures in Euclidean space

Abstract

Abstract. According to the size of sets for doubling measures, subsets of R n can be divided into six classes. Sets in these six classes are respectively called very thin, fairly thin, minimally thin, minimally fat, fairly fat, and very fat. In our main results, we prove that if a quasisymmetric mapping f of [0, 1] maps a uniform Cantor set E onto a uniform Cantor set f (E), then E is of positive Lebesgue measure if and only if f (E) is so. Also, we prove that the product of n uniform Cantor sets is very fat if and only if each of the factors is very fat, and that the product is minimally fat if and only if one of the factors is minimally fat.

A function whose graph has positive doubling measure

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We show that a doubling measure on the plane can give positive measure to the graph of a continuous function. This answers a question by Wang, Wen and Wen [3]. Moreover we show that the doubling constant of the measure can be chosen to be arbitrarily close to the doubling constant of the Lebesgue measure.

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On thin carpets for doubling measures

“…But for every d ≥ 2, there exist rectifiable curve in R d which is not thin, see [7]. In [27] the authors asked that: Is the graph of continuous function thin for doubling measures ? This question was negatived answered in [22].…”

On (alpha)_n-regular sets

“…Let us denote It is known that a symmetric Cantor set C(α n ) with defining sequence (α n ) is fat if and only if (α n ) ∈ ℓ 0 and thin if and only if (α n ) / ∈ ℓ ∞ (see [Wu93], [SW98] and [BHM12]). These results have also been generalized to nice (α n )-regular Cantor sets and uniform Cantor sets of the real line (see [CS12], [HWW09], [PW11] and [WWW13] for more precise definitions and results).…”

“…In literature fat sets have also been termed quasisymmetrically thick [Hei01], thick [HWW09] and very fat [BHM12,WWW13]. Thin sets, on the other hand, have also been called quasisymmetrically null [SW98], null for doubling measures [Wu93] and very thin [WWW13].…”

“…To prove Theorem 1.2, we need to prove four implications: [WWW13] that, the question of which sets are thin or fat in higher dimensional Euclidean spaces is still open. As a contribution related to this question, they show that a product of n uniform Cantor sets is fat if and only if each of the factors is fat and a product of n sets is thin if and only if some of the factors is thin.…”