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：2016
卷号：5
期号：12
页码：20394
DOI：10.15680/IJIRSET.2016.0512021
出版社：S&S Publications
摘要：In this paper we determine the estimates for upper / lower Box dimension of two dimensionalgeneralized Cantor like sets S. The two dimensional Cantor like sets S have been constructed in [3], for any sequence{ } with 0 < < 1 with the help of sequence of sets { } of subsets of [0, 1]× [0, 1] such that ⊃ , S =⋂ and m(S) = [Π ( 1 − )]. Further if Σ < ∞, m(S) ≠ 0 and if Σ = ∞, m(S) = 0.In particular it is proved that the upper and lower Box dimension of the set S satisfy () ≥→ { [ Π ( ) ] [ ] }and () ≤ → [Π ( ) ]. Further we generalize the propertyof sets of positive measure to Cantor like sets namely “ If T is a measurable set of positive measure then ∗ = { x – y : x∈ T , y ∈ T } = T – T contains an interval ( - , ) for some > 0”. We in fact prove that if S is Cantor like subset of[0, 1] then ∗ = S – S = [-1, 1].
关键词：Cantor set; Cantor like set; upper Box dimension and lower Box dimension.
