Fatness and thinness of uniform Cantor sets for doubling measures

Abstract

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.

Fat and thin sets for doubling measures in Euclidean space

“…See also Csörnyei and Suomala [2]. It plays an important role in proving the following result (see [1,6,13]). …”

On Cantor sets and doubling measures

“…After the submission of this paper, we were informed that for uniform Cantor sets, the result has been proved independently by Peng and Wen. See [PW11] for the precise formulation of their result.…”

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).…”