无穷数列集的Hausdorff测度和Hausdorff维数

Hausdorff Measure and Dimension of Sets of Sequence of Numbers

对于无穷数列集 R∞ ={z =(zi) ∞i=1:zi ∈R } ,定义度量 ρ(x ,y) =∑∞i=1｜xi- yi｜2 i(1+｜xi- yi｜) , x=(xi) ∞i=1、y=(yi) ∞i=1∈R∞ .在此度量下 ,考虑Hausdorff测度Hs,0≤s <∞ 。

We define the metric ρ(x,y)=∑∞i=1｜x i-y i｜2 i(1+｜x i-y i｜),for all x=(x i) ∞ i=1 ,y=(y i) ∞ i=1 ∈R ∞=｛z=(z i) ∞ i=1 :z i∈R,i≥1｝. We consider Hausdorff measure H s(0≤s<∞) by the metric ρ, and compute the Hausdorff dimension of some sets in R ∞.