期刊名称：International Journal of Innovative Research in Science, Engineering and Technology
印刷版ISSN：2347-6710
电子版ISSN：2319-8753
出版年度：2015
卷号：4
期号：12
页码：12270
DOI：10.15680/IJIRSET.2015.0412109
出版社：S&S Publications
摘要：The Cantor like sets S have been constructed in [3], for any sequence { 𝜖𝑛 } with 0 < 𝜖𝑛 < 1 with thehelp of sequence of sets { 𝑆𝑛 } of subsets of [ 0, 1 ] such that 𝑆𝑛 ⊃ 𝑆𝑛+1, S = 𝑆𝑛 and m(S) = ( 1 − ∞𝑛=1 𝜖𝑛 ).Further𝜖𝑛 = ∞ iff m(S) = 0.In this paper we determine the estimates for the Hausdorff and upper / lower Box dimension of such Cantorlike sets S. In particular it is proved that the Hausdorff dimension of the set S is given by− log 2𝛾, where 𝛾 =lim𝑘→∞ log[1− 𝜖𝑛2𝑘𝑛=1 ]1𝑘 and 𝑙𝑖𝑚𝑘→∞log 2−log [21− 𝜖 𝑛2𝑘𝑛=1 ]1𝑘≥ 𝑑𝑖𝑚𝐵(𝑆) ≥ 𝑑𝑖𝑚𝐵(𝑆) ≥𝑙𝑖𝑚𝑘→∞log 2−log { [1− 𝜖 𝑛2𝑘𝑛=1 ]1𝑘 [𝜖𝑘+1 ]1𝑘 }.If the above upper and lower limits are equal then 𝑑𝑖𝑚𝐵(𝑆) = 𝑑𝑖𝑚𝐵(𝑆) .
关键词：Cantor set; Cantor like set; Hausdorff dimension; upper Box dimension and lower Box dimension
