$\lambda$-power integrals on the Cantor type sets

Equilibrium Cantor-type Sets

Equilibrium Cantor-type sets are suggested. This allows to obtain Green functions with various moduli of continuity and compact sets with preassigned growth of Markov’s factors.

Dimensions of subsets of cantor-type sets

Let I = [0,1] be the unit interval on the real line andm> 1 be an integer. Let J = {0,1, . . . , m− 1}. For every point x ∈ I , there is a unique base-m representation x = Σk=1 jkm−k with jk ∈ J except for countable many points. Since countable sets do not interfere with our work, we neglect them here. For each j ∈ J , x ∈ [0,1], and n ∈ N, let τj(x,n) = {k : ik = j, 1 ≤ k ≤ n}, then the limit ...

Cantor sets

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ω of irrationals, or certain of its subspaces. In particular, given f ∈ (ω\{0}), we consider compact sets of the form Q i∈ω Bi, where |Bi| = f(i) for all, or for infinitely many, i. We also consider “n-splitting” compact sets, i.e., compact sets K such that for any f ∈ K and i ∈ ω, |{g(i) : ...

Two measures on Cantor sets

We give an example of Cantor-type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in the Stahl–Totik sense. c ⃝ 2014 Elsevier Inc. All rights reserved. MSC: 30C85; 31A15; 28A78; 28A80

On the Shape of Cantor Sets