Cantor三分集的一种新定义方式及其应用

A new definition method of Cantor trisection set and its application

通过对Cantor三分集本质结构进行深刻洞察,结合结构化方法对Cantor三分集的定义方式进行研究.定义了实数集上的无再生三分族及Cantor三分族等概念,提出Cantor三分族所对应的极限集就是传统所说的Cantor三分集的公设.具体的构造了一个Cantor三分族,并基于此给出了Cantor三分集的新公式P=3n-1+∞∩n=12∩i=1〔[0,2i-1/3n]∪[2i/3n,1]〕,利用所得公式证明了1/4属于Cantor三分集.这些研究结果使得Cantor三分集具有了更加自然、简捷、严密的公式表示,使得与Cantor三分集相关的问题的处理变得更加清晰、容易.

Through the deep insight into the essential structure of Cantor trisection set,and combined with the structured method,the definition of Cantor trisection set was further explored.The concepts of non reproducing trisection clan and Cantor trisection clan on real number set are defined,the postulate is proposed that the limit set corresponding to Cantor trisection clan is the traditionally called Cantor trisection set.A Cantor trisection clan was constructed specifically,and a new formula P=3n-1+∞∩n=12∩i=1〔[0,2i-1/3n]∪[2i/3n,1]〕for the Cantor trisection set was given based on this,the formula is used to prove that 1/4 belongs to Cantor trisection set.These results make Cantor trisection set have a more natural,simple and rigorous formula expression,and make the problems related to Cantor trisection set more clear and easy to deal with.