Dimensions of subsets of cantor-type sets

Fang-Xiong Zhen

doi:10.1155/ijmms/2006/26359

Abstract

We define the Cantor-type setEfirst, and then the Besicovitch subsetBpofE. We mainly show the dimensions of subsets of Cantor-type setEin compatible case and incompatible case.

Highlights

Let I = [0,1] be the unit interval on the real line and m > 1 be an integer
We mainly show the dimensions of subsets of Cantor-type set E in compatible case and incompatible case
We say that the Cantor-type set E satisfies the gap condition if there exists a constant δ > 0 such that dist I, I i1i2···in i1i2···in ≥ δ max

Summary

Introduction

For every point x ∈ I, there is a unique base-m representation x = Σ∞k=1 jkm−k with jk ∈ J except for countable many points. For each j ∈ J, x ∈ [0, 1], and n ∈ N, let τj(x, n) = {k : ik = j, 1 ≤ k ≤ n}, the limit τj(x) = limn→∞(1/n)τj(x, n) is called the frequency of number j in the base-m representation of x. A classical result of Borel [3] says that for Lebesgue almost every x ∈ [0,1], we have τj(x) = 1/m. Pm−1) such that Σj∈J pj = 1, consider the set. Pm−1) is composed of the number in [0, 1] having a ratio pj of digits equal to j in its base-m representation for each j.

Dimensions of subsets of cantor-type sets

Definitions and notations

Proofs